Assignment1
MTS26W2
(a) Let A be an $n \times n$ orthogonal matrix. Prove that the rows and columns
of A form an orthonormal set and that $A^T = A^{-1}$.
(b) Consider the subspaces $W_1$ and $W_2$ spanned by the vectors $v_1 = (1,1,0)^T$,
$v_2 = (0,1,1)^T$, and $w_1 = (1,0,1)^T$, $w_2 = (1,1,1)^T$ in $\mathbb{R}^3$. Find a vector
that lies in both $W_1$ and the orthogonal complement of $W_2$.
(c) Let A be a real symmetric matrix. Prove that the eigenspaces correspond-
ing to different eigenvalues of A are orthogonal.
