Nullspace, Column Space, and solution of system given only rref(A)

[nenna]

Homework Statement

Suppose a 3 x 5 matrix A has row-reduced echelon form:
[[1 2 0 0 5]
[0 0 1 0 4]
[0 0 0 1 3]]

a. Describe NS(A)
b. Describe CS(A)
c. Suppose
. [[2]
. [3] [[-2]
A [5] = [4] = b
. [1] [3]]
. [9]]

To be clear, that's the original matrix A times the vector x = {2, 3, 5, 1, 9} to give us the vector b = {-2, 4, 3}.

Find all solutions of the equation Ax = b

Homework Equations

The Attempt at a Solution

I completed a) by finding the basis for the nullspace to be the set of the two following vectors

{2, 1, 0, 0, 0} , {-5, 0, -4, -3, 1}

I'm not really sure how to do part b) since I'm not given the original matrix A, just the rref(A)

I have absolutely no idea how to do part c) since I'm not given the original matrix.

Please Help!

 

[mighty2000]

Hi there,

For b), you can still describe the column space of A even if you are only given the rref(A). Recall that the columns of the original matrix A are the same as the columns of the rref(A) that contain a leading 1. So in this case, the column space of A would be spanned by the first, third, and fourth column of the rref(A).

For c), you can use the fact that the solution to Ax = b is given by x = A^-1 * b. Since you are given the vector b, you can solve for x by finding the inverse of the original matrix A. The inverse of a matrix can be found by using the Gauss-Jordan elimination method.

Hope this helps! Let me know if you have any other questions.