Today

• row rank = column rank.

• Questions:

  • Q5 on the hw4. Part 2: Show that $\text{dim}(S_1 + S_2) \le \text{dim}(S_1) + \text{dim}(S_2)$. (Hint: show that basis vectors of $S_1$ together with basis vectors of $S_2$ span $S_1 + S_2$). Part 3: Show that $C(A+B) \subseteq C(A) + C(B)$.

  • Show how to prove column space and row space are vector spaces

  • Go over one example of QR factorization

  • Union of two subspaces $S_1 \cup S_2$ vs summing two subspaces together $S_1 + S_2$

  • Derivation of reflection of a matrix

  • A line/plane not passing 0, is it a vector space?

  • Review how to derive/prove Cauchy-Schwarz inequality

• Matrix inverse and linear equations (cont’d)