Linear Algebra (Basis)

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.

Thank you

Answer 1

(a) To solve the homogeneous system ( Ax = 0 ), where ( A^T * A = \begin{pmatrix} 3 & 4 & 0 \ 4 & 10 & -3 \ 0 & -3 & 3 \end{pmatrix} ), we need to find the null space of ( A ). This can be done by finding the eigenvectors corresponding to the eigenvalue 0 of ( A^T * A ). Then, the null space of ( A ) is spanned by these eigenvectors.(b) Given ( (A^T * A)^{-1} * A^T = \begin{pmatrix} \frac{3}{5} & -\frac{4}{5} & 0 \ \frac{-1}{5} & \frac{3}{5} & 0 \ \frac{-1}{5} & \frac{4}{15} & 0 \end{pmatrix} ), we can see that ( A = ((A^T * A)^{-1} * A^T)^T ). So, we transpose the given matrix to find ( A ).(c) To find all possible left inverses of ( A ), we look for matrices ( B ) such that ( BA = I ), where ( I ) is the identity matrix. We can find such ( B ) matrices by taking the inverse of ( A^T * A ) and multiplying it by ( A^T ).