Linear algebra calculator

Linear algebra calculator

What is basic linear algebra? Linear algebra refers to linear combinations. That is, use arithmetic on columns of numbers, called vectors, and arrays of numbers, called matrices, to create new columns and arrays of numbers. Linear algebra is the study of lines and planes, vector spaces and maps required for linear transformations.

What are the steps in solving linear equations?

Solving a linear equation: five steps to success.
Step 1 : Find the distribution for ().
Step 2 : Combine the same terms on each side of the = sign.
Step 3 : Add or subtract terms from variables so that all variables are on the same side of the = sign.

Is linear algebra the same as matrix algebra?

Linear algebra. Matrix algebra (matrix theory), matrix algebra is not the same as linear algebra, because matrix algebra can have a nonlinear function, matrix polynomial. Matrix analysis can have a derivative, but linear algebra cannot.

What is the formula for solving linear equations?

In mathematics, a linear equation is a type of equation. In a linear equation, two terms must be constant. A linear equation is an equation of a straight line. This type of equation is written in the form: y = mx + b OR (y y1) = m (x x1) OR m = rate of change or slope.

What are the prerequisites to studying linear algebra?

There are two major requirements for studying linear algebra in addition to regular high school courses that include algebra, geometry, and trigonometry (no, calculus is not a formal requirement). This is because you need to be able to obtain proof and know the basics of matrices and determinants.

What topics should be studied for linear algebra?

  • Mathematical operations with matrices (addition, multiplication)
  • Inverse and determinant matrices
  • Solving equation systems with matrices
  • Euclidean vector spaces
  • Eigenvalues ​​and Eigenvectors
  • Orthogonal matrices
  • Positive determined matrices
  • Linear Transformations
  • Projections
  • Linear dependence and independence

What is the difference between algebra and linear algebra?

Algebra is almost confused (as Steve said) with complex arithmetic. However, algebra simply refers to the manipulation of more abstract objects. Linear algebra refers to the algebraic manipulation of straight lines, vectors, scalars, systems of linear equations and matrices (bases).

:diamond_shape_with_a_dot_inside: What is the joy of learning linear algebra?

Original answer: What is the fun of learning linear algebra? The joy of discovering, seeing beauty, understanding logical thinking. In all sciences there is the joy of discovering what is, was and will be, something that is not an opinion or a matter of taste, it is not temporary.

:diamond_shape_with_a_dot_inside: What exactly is a basis in linear algebra?

In linear algebra, a basis is a collection of vectors in a given vector space with certain properties: any vector in a vector space can be obtained by multiplying each of the basis vectors by different numbers and then adding them.

What is advanced linear algebra?

Advanced linear algebra focuses on vector spaces and the maps between them that maintain their structure (linear transformations). It starts with familiar concepts and then gradually leads to deeper results.

What is basic linear algebra examples

The development of recommendation systems is mainly concerned with linear algebra methods. A simple example is to calculate the similarity between the scattered vectors of customer behavior using distance measures such as Euclidean distance or scalar products.

:brown_circle: What is basic linear algebra operations

Basic Linear Algebra Routines (BLAS) is a specification that prescribes a set of low-level routines for performing common linear algebra operations such as vector addition, scalar multiplication, scalar products, linear combinations, and matrix multiplication.

How important is linear algebra?

Linear algebra is important to engineers because it makes it easier to solve problems. Using matrices to solve a large system of equations greatly simplifies the process.

What does linear algebra stand for?

LA stands for Linear Algebra. The abbreviation is mainly used in the categories: Algebra Matrix Book Library Education.

Is there such thing as linear algebra?

Linear algebra, a mathematical discipline that deals with vectors and matrices and, more generally, vector spaces and linear transformations. Unlike other parts of mathematics, which are often inspired by new ideas and unsolved problems, linear algebra is very well studied.

:brown_circle: What is basic linear algebra exercises worksheets

Basic algebra is a very basic level of algebra in which the student learns to find the value of a variable. Each worksheet contains fifty basic algebraic problems so that students can practice algebraic expressions and learn to find the value of a variable.

Are there printable worksheets for solving linear equations?

Here you will find an unlimited number of printable worksheets for solving linear equations, available as PDF and HTML files. You can customize worksheets to include one-, two-, or multi-level equations, variables on both sides, parentheses, and more.

What do you need to know about algebra worksheets?

Print algebra worksheets. Algebra is a branch of mathematics that uses letters and symbols to represent numbers and quantities in formulas and equations.

What do the letters represent in an algebra worksheet?

Printable Algebra Worksheets Algebra is a branch of mathematics that uses letters and symbols to represent numbers and quantities in formulas and equations.

:diamond_shape_with_a_dot_inside: What is an expression in a algebraic equation?

An expression is a mathematical expression that combines variables and operations. What is an Algebraic Equation? In an equation, an algebraic expression equals something that can be a number or other algebraic expression.

What is basic linear algebra word problem

Solve word problems that lead to equations such as px + q = r and p(x + q) = r, where p, q and r are specific rational numbers. Solve the equations of these figures smoothly. Compare the algebraic solution with the arithmetic solution and determine the order of operations used in each method.

:diamond_shape_with_a_dot_inside: How do you solve word problems in math?

Usually, solving a word problem involves four simple steps: reading the problem and writing an equation using the words; O'CLOCK. an equation that contains both words and numbers. Whenever possible, use numbers instead of words to form a general math equation. Use math to solve the equation. Answer the question causing the problem.

What are examples of linear equations?

The definition of a linear equation is an algebraic equation in which each term has an exponent equal to unity and the graph of the equation is a straight line. An example of a linear equation is y = mx + b.

:brown_circle: What are basic linear algebra subprograms ( Blas )?

Basic Linear Algebra Routines (BLAS) is a specification that prescribes a set of low-level routines for performing common linear algebra operations such as vector addition, scalar multiplication, scalar products, linear combinations, and matrix multiplication.

When was the first linear algebra subprogram published?

A specification for these kernel operations using scalars and vectors was published in 1979, a Basic Linear Algebra Level 1 (BLAS) subroutine.

:eight_spoked_asterisk: What kind of loops are used in linear algebra subprograms?

The libraries will contain single and double precision versions of some algorithms. Originally, these routines used hard-coded loops for their low-level operations. For example, if a subroutine performs matrix multiplication, the subroutine has three nested loops.

Are there any low level routines for linear algebra?

These are de facto low-level standard routines for linear algebra libraries. The subroutines have links for C (CBLAS interface) and Fortran (BLAS interface). While the BLAS specification is generic, BLAS implementations are often optimized for speed on a particular computer, so using it can provide significant performance benefits.

:eight_spoked_asterisk: What are some examples of 2 Step equations?

However, solving two-step equations requires more than one mathematical step. An example of a two-step equation is 3x + 4 = 16. To solve this equation, first subtract 4 from both sides of the equation: 3x + 4 4 = 16 4. This gives the one-step equation 3x = 12.

:eight_spoked_asterisk: What are the rules for solving equations?

Solve equations and simplify expressions. In Algebra 1, they are taught that the two rules for solving equations are the addition rule and the multiplication/division rule. The addition rule for equations tells them that the same amount can be added to both sides of the equation without changing the set of solutions of the equation.

How do you calculate system of equations?

Solving by Multiplication Write one equation over another. Multiply one or both of the equations until one of the variables in the two terms has equal coefficients. Add or subtract equations. Solve until the end of the term. Plug the term back into the equation to find the value of the first term. Check your answer.

:eight_spoked_asterisk: What are the steps in solving linear equations by graphing

There are 4 steps to solve a linear system using a graph.
Step 1 : Specify the two equations as a slope.
Step 2 : Draw two equations on the same coordinate plane.
Step 3 : Estimate the intersection of the graphs.

What is the formula for graphing linear equations?

How to draw linear inequalities. A linear equation creates a straight line in the graph. The general formula for a linear equation is y = mx + b, where m is the slope of the line (which can be positive or negative) and b is the point where the line intersects the y-axis (the y-intercept ).

:diamond_shape_with_a_dot_inside: What is the rule for graphing linear equations?

1. Make sure that the linear equation is y = mx + b. It is known as the intersection form and is probably the easiest way to draw linear equations. The values ​​of the equations do not have to be whole numbers. You will often see an equation that looks like this: y = 1/4x + 5, where 1/4 is m and 5 is b.

What is the purpose of graphing linear equations?

Linear equation plotting is the process of placing equations with a linear pattern (positive or negative) on a line graph of the X and Y axes. An equation generally looks like y = mx + by representing the relationship between two points on a line..

What are the steps in solving linear equations by substitution

Solve systems of equations by substitution. Problem 1. If two equations are in the form of a slope segment, you can simply use substitution to solve the system by setting one value of y equal to the other value of y (since in both equations y equals something). From there, use inverse operations to find x.

How do you solve the linear system using substitution?

How to solve a system by replacement?
Step 1 : First solve the linear equation for y as a function of x.
Step 2 : Then replace that expression in another linear equation with y.
Step 3 : Solve this and you have the x-coordinate of the intersection.
Step 4 : Then enter x into one of the equations to find the corresponding y coordinate.

What is the system of equations using substitution?

Substitution is a method of solving a system of equations by eliminating all but one of the variables in one of the equations and then solving that equation. It does this by isolating another variable in one equation and then replacing the values ​​of those variables in another equation.

:eight_spoked_asterisk: What are the steps in solving linear equations and inequalities

The general procedure for solving multilevel inequalities is as follows. Remove the parentheses on both sides of the inequality and add like terms. Add or subtract terms so that the variable is on one side and the constant on the other side of the inequality sign. Multiply and divide by constants associated with a variable.

How do you solve one step inequality?

One-step inequality can be solved by adding once, subtracting once, multiplying once, or dividing once. If you multiply or divide an inequality by a negative number, the inequality sign should change direction.

How do you calculate an inequality?

Plug the slope and point into the formula y = mx + B, where m is the slope, (x, y) the point on the line and b is the y intersection to find an equation equal to the line -d -Definition of inequality. Substituting in (0, 0), you get = + b, so b = 0. If you rewrite the equation, you get y = x / 2.

:eight_spoked_asterisk: What are the steps in solving linear equations examples

Use four steps to solve the linear equation as follows: 1a. Scatter and combine similar terms. 1 B. Put similar terms next to each other and simplify. 2nd. Move the variables to the left of the equation. For this example, add -5x to each side of the equation.

What are the steps in solving linear equations by elimination

Let's see how they can use the elimination method to solve a system of equations. Step 1: Multiply or divide the linear equations by a number other than zero to get the total coefficient of one of the variables of the two equations, which will be numerically equal. Step 2: Then add or subtract the two equations to eliminate the same terms.

How can they solve systems of equations using elimination?

Elimination method to solve linear systems. Another way to solve a linear system is to use the method of elimination. Using elimination, addition, or subtraction equations to obtain an equation in one variable.

:diamond_shape_with_a_dot_inside: How do you solve linear equations with three variables?

Solve a system of three linear equations in three variables using Gaussian elimination. Divide the first equation by 3. Multiply by 4 and add 1 time to the second equation, then multiply by (1) and add to the third equation. they will get the following system: divide the second equation and get.

:diamond_shape_with_a_dot_inside: Is it possible to solve linear equation with two variables?

One way to solve a system of linear equations in two variables is to draw a graph. In this method, the equations are built on the same set of axes. Another method of solving a system of linear equations is substitution. In this method, you solve for a variable in one equation and enter the result in the second equation.

Does this system of linear equations have infinite solutions?

Systems of linear equations can have only 0, 1, or an infinite number of solutions. These two lines cannot be crossed twice. The correct answer is that the system has a solution.

What is the formula for solving linear equations in one variable

Any line graph is nothing more than a straight line. So if there are curves, it is not linear. Another way to find out is to look at your equation. If the equation can take the form Y = MX + B, where M and B are numbers, then it is a linear equation.

Can you give example of linear equation?

The definition of a linear equation is an algebraic equation in which each term has an exponent equal to unity and the graph of the equation is a straight line. An example of a linear equation is y = mx + b.

Can you have Division in a linear equation?

It is often necessary to use both multiplication and division to solve a linear equation. If the linear equation uses both multiplication and division, solve with the opposite operation for each.

What is the equation in one variable?

Equations in One Variable I. A equation in one variable is an equation of the form f(x) = g(x), where f(x) and g(x) are functions.

:brown_circle: Can you use a matrix in linear algebra?

However, it is more likely that linear algebra is talking about linear transformations that are not a list of numbers (I can't emphasize this), although it is sometimes useful to use a specific matrix to write the linear transformation.

:eight_spoked_asterisk: What's the difference between algebra and linear algebra?

However, algebra simply refers to the manipulation of more abstract objects. Linear algebra refers to the algebraic manipulation of straight lines, vectors, scalars, systems of linear equations and matrices (bases).

Which is the correct definition of matrix algebra?

What you call matrix algebra are actually properties of linear maps for finite-dimensional vector spaces. Linear algebra in its most general definition has to do with finite and infinite dimensions.

:brown_circle: Which is the branch of mathematics concerned with linear equations?

Linear algebra is a branch of mathematics that deals with linear equations such as: and their representations in vector spaces and in terms of matrices. Linear algebra is at the heart of almost all areas of mathematics.

Is linear algebra the same as matrix algebra worksheet

What you call matrix algebra are actually linear map properties for finite-dimensional vector spaces. Linear algebra in its most general definition deals with both finite and infinite dimensions. If it is a finite dimensional linear algebra problem (two dimensional because it has two variables), it can be represented by matrices.

Which is the inverse of a matrix algebra?

In matrix algebra, an inverse is a matrix that, when multiplied by the original matrix, gives the identity matrix. The inverse matrix is ​​indicated by the indicator "1". A matrix must be square to have an inversion, but not all square matrices have an inversion.

:diamond_shape_with_a_dot_inside: How are matrices added and subtracted in matrix algebra?

Matrix addition and subtraction: To add two matrices, they must have the same number of rows and the same number of columns. The elements of the two arrays are simply added element by element to get the results.

Which is a column vector in matrix algebra?

It is important to keep this convention in mind when performing matrix algebra. A vector is a special type of matrix that contains only one row (called a row vector) or one column (called a column vector). Below a is a column vector and b is a row vector.

:diamond_shape_with_a_dot_inside: Is linear algebra the same as matrix algebra 1

Matrix theory is a specialization of linear algebra in the case of finite-dimensional vector spaces and through explicit manipulations after establishing a basis. Sometimes specific calculations are needed using a matrix view.

Why is linear algebra important in math and science?

Linear algebra is a branch of mathematics that deals with linear equations such as: and their representations in vector spaces and in terms of matrices. Linear algebra is also a foundation for numerical calculus that works with vectors and matrices for all calculations. Linear algebra is at the heart of almost all areas of mathematics.

:eight_spoked_asterisk: What are the applications of matrix algebra in geometry?

These "matrix transformations" are an important tool in geometry, and geometry in turn provides a "picture" of the matrices. In addition, matrix algebra has many other applications, some of which are discussed in this chapter.

:eight_spoked_asterisk: Is linear algebra the same as matrix algebra examples

In this chapter you look at the matrices themselves. While some of the motivation comes from linear equations, it turns out that matrices can be multiplied and added to form an algebraic system somewhat similar to real numbers. This "algebra matrix" is not as useful as studying linear equations.

What's the difference between linear algebra and regular algebra?

Mathematics and Physics, University of Houston, Clear Lake (2007) Algebra itself is the study and definition of unknowns. There are many kinds of unknowns and many ways to study them, hence the many variants of algebra. "Regular" algebra is the study of normal, everyday "real" numbers. Linear algebra is the study of matrices.

When do two matrices have the same size?

When a record is highlighted, the first index refers to the row and the second to the column in which it is located. Two points and in a plane are absolutely equal if they have the same coordinates, that is, and. Likewise, two matrices are said to be equal (written) only if: They have the same size.

:eight_spoked_asterisk: When is a system of linear equations consistent?

1 Any system of linear equations has the form in which the coefficient matrix, the constant matrix and the variable matrix are. 2 A system is consistent if and only if it is a linear combination of columns. 3 If they are columns and if, then the solution of the linear system is a solution of the vector equation if and only if.

:brown_circle: Which is an example of a matrix transformation?

For example, the geometric transformations obtained by rotating the Euclidean plane around the origin can be seen as a multiplication by certain matrices. These "matrix transformations" are an important tool in geometry, and geometry in turn provides a "picture" of the matrices.

:brown_circle: When are matrices A and B inverses of each other?

Matrices A and B are inverse if and only if AB and BA are identity matrices. An invertible matrix has an inverse. Let A be a matrix of R x C. Then A is a linear algebra function (set) if and only if | R | = | C | and the columns of matrix A are linearly independent.

:eight_spoked_asterisk: What kind of problems can linear algebra be used for?

Linear algebra deals with structures that can be described by systems of linear equations (that is, equations containing only the first powers of all variables). Typical applications are certain types of optimization problems (linear programming) and vector space exploration.

What are the steps to solve linear equations?

Solve linear equations.
Step 1. Eliminate fractions or decimals.
Step 2. Simplify any part of the equation by removing parentheses and combining like terms.
Step 3. Highlight the variable term on one side of the equation.
Step 4. Solve the equation by dividing each side of the equation.
Step 5. Check your solution.

:diamond_shape_with_a_dot_inside: How do I create a linear equation?

Steps Make sure the linear equation is y = mx + b. Imagine the number b on the Y-axis. Convert m to a fraction. Continue the line from point b, starting on the slope or uphill. Continue to extend the line using a ruler, remembering to use the slope m as a guideline.

What are the rules of linear equations?

Three basic rules. A linear equation consists of two expressions (for example, "3x + 2" or "54") that are equal to each other, provided neither variable in the equation is raised to a power greater than one.

:eight_spoked_asterisk: What is the formula for solving linear equations by graphing

To solve a system of equations using a graph, draw each equation and determine the intersection of the two lines. This point is the solution of the system of equations, here the x and y values ​​of the two equations coincide. Check the solution by substituting the values ​​of each equation.

:diamond_shape_with_a_dot_inside: Can you solve system of equations by graphing?

Solve systems of equations using graphs. A system of linear equations contains two or more equations y = + 2 and y = x2. The solution of such a system is an ordered pair, which is the solution of both equations. To graphically solve a system of linear equations, draw both equations in the same coordinate system.

:brown_circle: What is the formula for solving linear equations worksheet

Solving a Linear Equation Simplify both sides of the equation by multiplying and collecting common terms for variables and constants separately. Eliminate the variable 8x to the right by subtracting 8x from both sides. Eliminate the constant term 22 from the left by subtracting 22 from both sides.

What is a simple linear equation?

Linear Equations A simple linear equation is: y = mx + c A linear equation graphically looks like a straight line. It has a constant slope value. The degree of a linear equation is always equal to 1. The superposition principle applies to a system characterized by a linear equation.

How do you solve linear inequalities?

To solve a linear inequality, you need to find all the combinations of x and y that make the inequality true. You can solve linear inequalities using algebra or graphing. To solve a linear inequality (or any other equation), you need to find all the combinations of x and y that make that equation true.

How do you solve system of equations using substitution?

Replacement method. One way to solve systems of equations is by substitution. In this method, you solve an equation for one variable, replace that solution with another equation, and then solve it.

How many solutions do two linear inequalities have?

Linear inequalities can have no solution, a specific solution, or an infinite number of solutions. So the possible total would be three. For example, let's say you have a variable x.

What are examples of solving equations?

  • Example 1: Solution x 3 = for x gives the solution x = 3
  • Example 2: Solution x5 = for x gives the solution x = 5
  • Example 3. The solution x²1 = 0 with respect to x leads to two valid solutions: x = 1 and x = 1.
  • Example 4: Solving x y = for y gives y = x.
  • Example 5: The solution x y + 2 = for y results in the solution y = x + 2.

:diamond_shape_with_a_dot_inside: What is the formula for solving linear equations by elimination

The elimination method for solving systems of linear equations uses the addition property of equality. You can add the same value to both sides of the equation. So if you have a system: x - 6 = -6 and x + y = 8, then you can add x + y to the left side of the first equation and 8 to the right side of the equation.

:eight_spoked_asterisk: How do you solve the system of linear equations?

There are three ways to solve a system of linear equations: tracing, substitution, and elimination. The solution of a system of linear equations is an ordered pair (or pairs) that satisfies all the equations in the system.

How do I solve by elimination?

The steps to solve equations using the elimination method are as follows:
Step 1 : Rewrite the system of equations and combine it with the unknowns.
Step 2 : Exploring the unknown has opposite coefficients. If yes, add the two equations, the result is a one-variable equation.

How do I solve algebra?

Solve the two-step equation by multiplying instead of dividing at the end. The principle for solving this type of equation is the same: use arithmetic to add constants together, isolate a variable member, and then isolate the variable without isolating the member. Suppose you are working with the equation x / 5 + 7 = 3.

What are the steps in Algebra?

Steps of the solution method. the word problem in algebra is as follows.
STEP 1 : Read the problem carefully and write it down. what is given and what is needed.
STEP 2 : Select a letter or letters for x (and y). Unknown quantities requested.

How do you calculate variables in Algebra?

Algebraic equations are usually written with numbers and/or variables on both sides, as shown below: x + 2 = 9 × 4. To find a variable, you only need to place it on one side of the equals sign. On the other side of the equal sign, only your answer remains.

:brown_circle: What are the steps to solve a math problem?

5 Steps to Solve Problems in a Real Math Drama
Step #1 : Stop and think before you act.
Step #2 : Translate from English to Comparison
Step #3 : Decide what interests you
Step #4 : Make sure you understand the result
Step #5 : Use your result to solve other problems.

:diamond_shape_with_a_dot_inside: How does the algebra calculator with steps work?

The algebra calculator can help you find solutions to a wide variety of math problems. The calculator works with both equations and expressions. Basically, calculus solves the following algebraic problems: finding unknowns, calculating, fractions, quadratic equations, simplification, factoring, etc. This is how the step math algebra calculator works.

:eight_spoked_asterisk: Do you need to use an algebra calculator?

When you need to solve algebra tasks or help your child solve a difficult problem or equation, you can use the Math Algebra Calculator, an application that helps students understand their mistakes and overcome them continuously. practice. First, why would you want to study algebra?

What do you need to know about the equation calc?

Basically, calculus solves the following algebraic problems: finding unknowns, calculus, fractions, quadratic equations, simplification, factoring, etc. Math likes simplicity and the calculator is simple and effective to use. The following steps describe the basic use of your online algebra calculator:.

What are the symbols in an algebra calculator?

Math symbols. If you want to create your own math expressions, the calculator includes some symbols here: + (addition) (subtraction) * (multiplication).

:brown_circle: What website can solve any math problem?

This article covers 5 math problem solving websites: Mathway, WolframAlpha, WebMath, Solve My Math, and Tiger Algebra. Mathway is the first website dedicated to solving math problems. To use this site, you must first create a free account. Once inside, you can report your concerns in the designated area.

How do you solve equations step by step?

Solve linear equations.
Step 1. Eliminate fractions or decimals.
Step 2. Simplify any part of the equation by removing parentheses and combining like terms.
Step 3. Highlight the variable term on one side of the equation.
Step 4. Solve the equation by dividing each side of the equation.

How do I solve an unknown equation?

An equation of the form: ax + b = 0, where a and b are known numbers, x is an unknown quantity, is called a linear equation with one unknown. Solving this equation means finding the numerical value x for which this equation becomes an identity.

How do you solve the matrix equation?

Matrix Equation Solution Standardize your matrices so that they can be used in the standard matrix equation form Ax = B. For this set of instructions, the matrix equation x = can be used to find the process solution of the equation to illustrate. Create matrix A. Create matrix B.

:eight_spoked_asterisk: How do you calculate the inverse of a matrix?

You can calculate the inverse of the matrix:
Step 1 : Calculation of the matrix of minors,
Step 2 : then convert this to a cofactor matrix,
Step 3 : than a drug addict, and.
Step 4 : multiply this by 1/determinant.

:diamond_shape_with_a_dot_inside: How to calculate in a matrix?

Make sure the number of columns in the 1st is equal to the number of rows in the 2nd. Multiply the elements of each row of the first matrix by the elements of each column of the second matrix. Add products.

How do you calculate the determinant of a matrix?

Find the determinant. Type 3 x 3. Select a row or column. Cross out the row and column of your first article. Find the determinant of the 2 x 2 matrix and multiply the answer by the selected element. Determine the sign of your answer. Repeat this process for the second item in the referenced row or column.

What order to multiply matrices?

Matrix multiplication is done in SRT order, where S, R and T are matrices for scaling, rotation and translation, respectively. The order of a compound transformation is first scaled, then rotated, and then moved.

How do you find the range of a matrix?

Solution. By definition, the interval R(A) of matrix A is equal to R(A) = {b ∈ R3 | Ax = b for some x ∈ R4}. The vector b = in R3 lies in the interval R (A) if and only if the system Ax = b is consistent. So let's find the conditions for b so that the system is consistent.

linear algebra calculator