Suppose the system AX = B is consistent and A is a 6x3 matrix. Suppose the...

Suppose the system AX = B is consistent and A is a 6x3 matrix. Suppose the maximum number of linearly independent rows in A is 3. Discuss: Is the solution of the system unique?

Solutions

Expert Solution

Since the system is consistent, there is at least one solution, say . Let be any solution to , and let ; then

Thus, .

Since the maximum number of linearly independent rows in is , we know that is the row-rank of . Since row-rank and column-rank of a matrix are identical, we conclude that column-rank of is also . Thus, all the columns of are linearly independent. Since the vector is a linear combination of the columns of , namely, , where and are columns of , we conclude that implies . This implies

Thus, if are solutions to then ; hence, the system has a unique solution.

Colby Messinger answered 1 year ago

Next > < Previous

Related Solutions

A linear system of equations Ax=b is known, where A is a matrix of m by...

A linear system of equations Ax=b is known, where A is a matrix of m by n size, and the column vectors of A are linearly independent of each other. Please answer the following questions based on this assumption, please explain all questions, thank you~. (1) Please explain why this system has at most one solution. (2) To give an example, Ax=b is no solution. (3) According to the previous question, what kind of inference can be made to the...

Solve the linear system of equations Ax = b, and give the rank of the matrix...

Solve the linear system of equations Ax = b, and give the rank of the matrix A, where A = 1 1 1 -1 0 1 0 2 1 0 1 1 b = 1 2 3

for the system equation of x' = Ax if Coefficients Matrix A be ? = [...

for the system equation of x' = Ax if Coefficients Matrix A be ? = [ 5 −5 −5 −1 4 2 3 −5 −3 ] , find the basic matrix

Suppose A is an n × n matrix with the property that the equation Ax =...

Suppose A is an n × n matrix with the property that the equation Ax = b has at least one solution for each b in R n . Explain why each equation Ax = b has in fact exactly one solution

Let A be an m x n matrix. Prove that Ax = b has at least...

Let A be an m x n matrix. Prove that Ax = b has at least one solution for any b if and only if A has linearly independent rows. Let V be a vector space with dimension 3, and let V = span(u, v, w). Prove that u, v, w are linearly independent (in other words, you are being asked to show that u, v, w form a basis for V)

If an SPL ( LINEAR EQUATION SYSTEM ) is known: Ax = b. A is a...

If an SPL ( LINEAR EQUATION SYSTEM ) is known: Ax = b. A is a matrix sized m × n and b is a vector sized m × 1, with the component values of matrix A and vector b known. The x vector is n × 1 and the component values are unknown. Explain how the possible solution of SPL Ax = b. i want answer for the question , and what he mean by (the component values of...

1) Let A be nxn matrix and Ax=b, if we need change A to Upper triangular...

1) Let A be nxn matrix and Ax=b, if we need change A to Upper triangular matrix using Gaussian Elimination, how many additions/subtraction operations are involved? how many multiplication/division operations are involved? 2) Once we got the upper triangular matrix, now we need to apply back-substitution, how many additions/subtraction operations are involved? how many multiplication/division operations are involved?

Given the differential equation (ax+b)2d2y/dx2+(ax+b)dy/dx+y=Q(x) show that the equations ax+b=et and t=ln(ax+b) reduces this equation to...

Given the differential equation (ax+b)2d2y/dx2+(ax+b)dy/dx+y=Q(x) show that the equations ax+b=et and t=ln(ax+b) reduces this equation to a linear equation with constant coefficients hence solve (1+x)2d2y/dx2+(1+x)dy /dx+ y=(2x+3)(2x+4)

Matrix: Ax b [2 1 0 0 0 | 100] [1 1 -1 0 -1 |...

Matrix: Ax b [2 1 0 0 0 | 100] [1 1 -1 0 -1 | 0] [-1 0 1 0 1 | 50] [0 -1 0 1 1 | 120] [0 1 1 -1 1 | 0] Problem 5 Compute the solution to the original system of equations by transforming y into x, i.e., compute x = inv(U)y. Solution: %code I have not Idea how to do this. Please HELP!

For matrices, a mulitplicative identity is a square matrix X such XA = AX = A...

For matrices, a mulitplicative identity is a square matrix X such XA = AX = A for any square matrix A. Prove that X must be the identity matrix. Prove that for any invertible matrix A, the inverse matrix must be unique. Hint: Assume that there are two inverses and then show that they much in fact be the same matrix. Prove Theorem which shows that Gauss-Jordan Elimination produces the inverse matrix for any invertible matrix A. Your proof cannot...

Subjects