Briefly describe the use of Gaussian elimination in solving cadi of the followin

Question

Briefly describe the use of Gaussian elimination in solving cadi of the following problems. State and prove the basic principle behind why the procedure works. (Example: To find the inverse of a matrix: Construct the augmented matrix [M/I]. Reduce it to [I|N]. Then N is the inverse of M. This works because when you row reduce you are in fact multiplying on the left by an invertible matrix Q. So [IN] = Q[M|I] which is [QM|Q]. So QM = I and N = Q.) Find a basis for the column space of a matrix. Find a basis for the nullspace of a matrix. Find a basis for the rows pace of a matrix. Given a matrix M, find another matrix N such that the nullspace of N is the column space of M.

Explanation / Answer

i) To find the basis for the column space of a matrix A, We first take the transpose of this matrix A,

Then this matrix is reduced to row-echlon form by Gaussian elimination

The non zero rows in the transpose of A, are the linearly independent columns of A.

ii) To find the basis of nullspace of A we have to solve AX=0

The matrix A is reduced to row-echlon form by Gaussian elimination.

And the non-zero rows are the required basis for the nullspace

iii) To fnd the basis of rowspace of A

The matrix A is reduced to row-echlon form by Gaussian elimination.

And the non-zero rows are the required basis for the rowspace.

iv) We take any matrix A, And reduce it to row-echlon form.

The number of linearly independent rows is equal to the number of linearly oindependent columns.

Taking A=N and M=Tran(A)

We have the required matrices

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