Sums and Scalar Products

Perform the indicated operations, if defined, or state “not defined.”

• $\begin{pmatrix}5&-1&2\\ 6&1&1\end{pmatrix}+\begin{pmatrix}2&1&4\\ 3&0&5\end{pmatrix}$

• $6\cdot\begin{pmatrix}2&-1&-1\\ 1&2&3\end{pmatrix}$

• $\begin{pmatrix}2&1\\ 0&3\end{pmatrix}+\begin{pmatrix}2&1\\ 0&3\end{pmatrix}$

• $4\begin{pmatrix}1&2\\ 3&-1\end{pmatrix}+5\begin{pmatrix}-1&4\\ -2&1\end{pmatrix}$

• $3\begin{pmatrix}2&1\\ 3&0\end{pmatrix}+2\begin{pmatrix}1&1&4\\ 3&0&5\end{pmatrix}$

Give the matrix representing the zero map from $\mathbb{R}^{4}$ to $\mathbb{R}^{2}$, with respect to the standard bases.

Prove Theorem 1.4.

• Prove that matrix addition represents addition of linear maps.

• Prove that matrix scalar multiplication represents scalar multiplication of linear maps.

Prove each, assuming that the operations are defined, where $G$, $H$, and $J$ are matrices, where $Z$ is the zero matrix, and where $r$ and $s$ are scalars.

• Matrix addition is commutative $G+H=H+G$.

• Matrix addition is associative $G+(H+J)=(G+H)+J$.

• The zero matrix is an additive identity $G+Z=G$.

• $0\cdot G=Z$

• $(r+s)G=rG+sG$

• Matrices have an additive inverse $G+(-1)\cdot G=Z$.

• $r(G+H)=rG+rH$

• $(rs)G=r(sG)$

Fix domain and codomain spaces. In general, one matrix can represent many different maps with respect to different bases. However, prove that a zero matrix represents only a zero map. Are there other such matrices?

Let $V$ and $W$ be vector spaces of dimensions $n$ and $m$. Show that the space $\mathop{{\cal L}}(V,W)$ of linear maps from $V$ to $W$ is isomorphic to $\mathcal{M}_{m\!\times\!n}$.

Show that it follows from the prior question that for any six transformations $t_{1},\dots,t_{6}\colon\mathbb{R}^{2}\to\mathbb{R}^{2}$ there are scalars $c_{1},\dots,c_{6}\in\mathbb{R}$ such that not every $c_{i}$ equals $0$ but $c_{1}t_{1}+\dots+c_{6}t_{6}$ is the zero map. (Hint: the six is slightly misleading.)

The trace of a square matrix is the sum of the entries on the main diagonal (the $1,1$ entry plus the $2,2$ entry, etc.; we will see the significance of the trace in Chapter Five). Show that $\text{trace}(H+G)=\text{trace}(H)+\text{trace}(G)$. Is there a similar result for scalar multiplication?

Recall that the transpose of a matrix $M$ is another matrix, whose $i,j$ entry is the $j,i$ entry of $M$. Verify these identities.

• ${{(G+H)}^{\mathsf{T}}}={{G}^{\mathsf{T}}}+{{H}^{\mathsf{T}}}$

• ${{(r\cdot H)}^{\mathsf{T}}}=r\cdot{{H}^{\mathsf{T}}}$

A square matrix is symmetric if each $i,j$ entry equals the $j,i$ entry, that is, if the matrix equals its transpose.

• Prove that for any square $H$, the matrix $H+{{H}^{\mathsf{T}}}$ is symmetric. Does every symmetric matrix have this form?

• Prove that the set of $n\!\times\!n$ symmetric matrices is a subspace of $\mathcal{M}_{n\!\times\!n}$.

• How does matrix rank interact with scalar multiplication — can a scalar product of a rank $n$ matrix have rank less than $n$? Greater?

• How does matrix rank interact with matrix addition — can a sum of rank $n$ matrices have rank less than $n$? Greater?