Linear Algebra Basis/Subspaces Invertibility

Question

Given A^T * A = (3 4 0; 4 10 -3; 0 -3 3).

(a) Solve the homogenous system Ax = 0. Explain how you derive your answer.

(b) Suppose (A^T * A)^-1 * A^T = (3/5 -4/5 0 3/5 1/5; -1/5 3/5 0 -1/5 -2/5; -1/5 4/15 0 -8/15 -11/15). What is A? Explain how you derive your answer.

(c) Find all possible left inverse of A. Show your working clearly, however, you do not need to show the elementary row operations used.

Thanks

Answer 1

I won't give you the answers but here's how you would go about solving them:

(a) Either use a system of equations or row reduce the augmented matrix. Check out this handy article if you're not familiar with these approaches: https://textbooks.math.gatech.edu/ila/row-reduction.html

(b) This one's longer but you would define some unknown matrix as A - so let A = (x1, x2, x3, x4, x5; y1, y2, y3, y4, y5; z1, z2, z3, z4, z5) and plug it into the given equation and simplify to find values of xi, yi, and zi (where i = 1, 2, 3, 4, 5).

(c) Hint: For an MxN matrix A, if M>N, the left-inverse of A is given by (A^T A)^-1 A^T. Alternatively, you might consider doing something similar to (b) above wherein you let A^-1 equal some unknown matrix and find the values such that A^-1 A = Identity.

Hope this helps!

- Amaan.