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Linear Algebra

Homework 1 - Practice with matrix algebra

Due 2021-08-31. Remember thinking like a mathematician involving providing justification for each conclusion or formula. In the language of Math 300, a proof.

For this assignment, you can use the following fact about matrix addition and multiplication: distributivity. \(A(B+C) = AB + AC\) Remember that $AC \neq CA$ in general for matrices. (A good mental exercise is to think about how you would, as a good mathematician, prove distributivity of matrix multiplication over addition.)

  1. (Do this individually) On the channel in Microsoft Teams, introduce yourself. Also, tell us why you are taking the course and what you hope to gain from it.

  2. One of the most important matrices is the $m \times m$ identity matrix. We denote it by $I_m$. It’s entries are \(I_{i,j} = \begin{cases} 0 & \text{ if } i \neq j \\ 1 & \text{ if } i = j \end{cases}\) For example, the $3 \times 3$ identity matrix is \(\begin{pmatrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}\) Let $A$ be any $m \times n$ matrix and let $B$ be any $s \times m$ matrix. What are
    • $ I A $?
    • $ B I $?

  3. For a scalar $c$ and two integers $1 \leq l \leq m$, we let $S(l,c)_m$ denote the the $m \times m$ matrix whose entries are \(\left( S(l,c)_m \right)_{i,j} = \begin{cases} 0 & \text{ if } i \neq j \\ 1 & \text{ if } i = j \neq l \\ c & \text{ if } i = j = l \end{cases}\) For an $m \times n$ matrix $A = (a_{ij})$, give formulas for the entries of the product $S(l,c) A$. (Hint: Try to do the case $m = 2$ and see if you can see a pattern. Test $m = 3$ if you need more data.)

  4. Recall the matrices $D(i,j)_m$. Show that, for any $m \times n$ matrix $A$, the product $D(i,j)A$ is zero except for the i-th row which is equal to the $\mathbf{R}_j^A$, the j-th row of $A$.

  5. For the permutation matrices $P(i,j)_m$, verify that the identity \(P(i,j) P(r,s) = P(r,s) P(i,j)\) is true if \(\lbrace i,j \rbrace \cap \lbrace r,s \rbrace = \empty\) What happens in the case that just one of the indices are the same? Both?