# Linear Algebra/Sums and Scalar Products

Recall that for two maps $f$ and $g$ with the same domain and codomain, the map sum $f+g$ has this definition.
    ${\vec {v}}\;{\stackrel {f+g}{\longmapsto }}\;f({\vec {v}})+g({\vec {v}})$
The easiest way to see how the representations of the maps combine to represent the map sum is with an example.

## Example 1.1

Suppose that $f,g:\mathbb {R} ^{2}\to \mathbb {R} ^{3}$ are represented with respect to the bases $B$ and $D$ by these matrices.
    $F={\rm {Rep}}_{B,D}(f)={\begin{pmatrix}1&3\\2&0\\1&0\end{pmatrix}}_{B,D}\qquad G={\rm {Rep}}_{B,D}(g)={\begin{pmatrix}0&0\\-1&-2\\2&4\end{pmatrix}}_{B,D}$
Then, for any ${\vec {v}}\in V$ represented with respect to $B$, computation of the representation of $f({\vec {v}})+g({\vec {v}})$
    ${\begin{pmatrix}1&3\\2&0\\1&0\end{pmatrix}}{\begin{pmatrix}v_{1}\\v_{2}\end{pmatrix}}+{\begin{pmatrix}0&0\\-1&-2\\2&4\end{pmatrix}}{\begin{pmatrix}v_{1}\\v_{2}\end{pmatrix}}={\begin{pmatrix}1v_{1}+3v_{2}\\2v_{1}+0v_{2}\\1v_{1}+0v_{2}\end{pmatrix}}+{\begin{pmatrix}0v_{1}+0v_{2}\\-1v_{1}-2v_{2}\\2v_{1}+4v_{2}\end{pmatrix}}$
gives this representation of $f+g\,({\vec {v}})$.
    ${\begin{pmatrix}(1+0)v_{1}+(3+0)v_{2}\\(2-1)v_{1}+(0-2)v_{2}\\(1+2)v_{1}+(0+4)v_{2}\end{pmatrix}}={\begin{pmatrix}1v_{1}+3v_{2}\\1v_{1}-2v_{2}\\3v_{1}+4v_{2}\end{pmatrix}}$
Thus, the action of $f+g$ is described by this matrix-vector product.
    ${\begin{pmatrix}1&3\\1&-2\\3&4\end{pmatrix}}_{B,D}{\begin{pmatrix}v_{1}\\v_{2}\end{pmatrix}}_{B}={\begin{pmatrix}1v_{1}+3v_{2}\\1v_{1}-2v_{2}\\3v_{1}+4v_{2}\end{pmatrix}}_{D}$
This matrix is the entry-by-entry sum of original matrices, e.g., the $1,1$ entry of ${\rm {Rep}}_{B,D}(f+g)$ is the sum of the $1,1$ entry of $F$ and the $1,1$ entry of $G$.
Representing a scalar multiple of a map works the same way.

## Example 1.2

If $t$ is a transformation represented by
    ${\rm {Rep}}_{B,D}(t)={\begin{pmatrix}1&0\\1&1\end{pmatrix}}_{B,D}\quad {\text{so that}}\quad {\vec {v}}={\begin{pmatrix}v_{1}\\v_{2}\end{pmatrix}}_{B}\mapsto {\begin{pmatrix}v_{1}\\v_{1}+v_{2}\end{pmatrix}}_{D}=t({\vec {v}})$
then the scalar multiple map $5t$ acts in this way.
    ${\vec {v}}={\begin{pmatrix}v_{1}\\v_{2}\end{pmatrix}}_{B}\;\longmapsto \;{\begin{pmatrix}5v_{1}\\5v_{1}+5v_{2}\end{pmatrix}}_{D}=5\cdot t({\vec {v}})$
Therefore, this is the matrix representing $5t$.
    ${\rm {Rep}}_{B,D}(5t)={\begin{pmatrix}5&0\\5&5\end{pmatrix}}_{B,D}$

## Definition 1.3

The sum of two same-sized matrices is their entry-by-entry sum. The scalar multiple of a matrix is the result of entry-by-entry scalar multiplication.

## Remark 1.4

These extend the vector addition and scalar multiplication operations that we defined in the first chapter.

## Theorem 1.5

Let $h,g:V\to W$ be linear maps represented with respect to bases $B,D$ by the matrices $H$ and $G$, and let $r$ be a scalar. Then the map $h+g:V\to W$ is represented with respect to $B,D$ by $H+G$, and the map $r\cdot h:V\to W$ is represented with respect to $B,D$ by $rH$.

## Proof

Problem 2; generalize the examples above.
A notable special case of scalar multiplication is multiplication by zero. For any map $0\cdot h$ is the zero homomorphism and for any matrix $0\cdot H$ is the zero matrix.

## Example 1.6

The zero map from any three-dimensional space to any two-dimensional space is represented by the $2\!\times \!3$ zero matrix
    $Z={\begin{pmatrix}0&0&0\\0&0&0\end{pmatrix}}$
no matter which domain and codomain bases are used.

## Exercises

    This exercise is recommended for all readers.

### Problem 1

Perform the indicated operations, if defined.
* ${\begin{pmatrix}5&-1&2\\6&1&1\end{pmatrix}}+{\begin{pmatrix}2&1&4\\3&0&5\end{pmatrix}}$
* $6\cdot {\begin{pmatrix}2&-1&-1\\1&2&3\end{pmatrix}}$
* ${\begin{pmatrix}2&1\\0&3\end{pmatrix}}+{\begin{pmatrix}2&1\\0&3\end{pmatrix}}$
* $4{\begin{pmatrix}1&2\\3&-1\end{pmatrix}}+5{\begin{pmatrix}-1&4\\-2&1\end{pmatrix}}$
* $3{\begin{pmatrix}2&1\\3&0\end{pmatrix}}+2{\begin{pmatrix}1&1&4\\3&0&5\end{pmatrix}}$

### Problem 2

Prove Theorem 1.5.
* Prove that matrix addition represents addition of linear maps.
* Prove that matrix scalar multiplication represents scalar multiplication of linear maps.
    This exercise is recommended for all readers.

### Problem 3

Prove each, where 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 $G+(-1)\cdot G=Z$.
* $r(G+H)=rG+rH$
* $(rs)G=r(sG)$

### Problem 4

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?
    This exercise is recommended for all readers.

### Problem 5

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

### Problem 6

Show that it follows from the prior questions that for any six transformations $t_{1},\dots ,t_{6}:\mathbb {R} ^{2}\to \mathbb {R} ^{2}$ there are scalars $c_{1},\dots ,c_{6}\in \mathbb {R}$ such that $c_{1}t_{1}+\dots +c_{6}t_{6}$ is the zero map. (Hint: this is a bit of a misleading question.)

### Problem 7

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 ${\mbox{trace}}(H+G)={\mbox{trace}}(H)+{\mbox{trace}}(G)$. Is there a similar result for scalar multiplication?

### Problem 8

Recall that the transpose of a matrix $M$ is another matrix, whose $i,j$ entry is the $j,i$ entry of $M$. Verifiy these identities.
* ${{(G+H)}^{\rm {trans}}}={{G}^{\rm {trans}}}+{{H}^{\rm {trans}}}$
* ${{(r\cdot H)}^{\rm {trans}}}=r\cdot {{H}^{\rm {trans}}}$
    This exercise is recommended for all readers.

### Problem 9

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 $H$, the matrix $H+{{H}^{\rm {trans}}}$ is symmetric. Does every symmetric matrix have this form?
* Prove that the set of $n\!\times \!n$ symmetric matrices is a subspace of ${\mathcal {M}}_{n\!\times \!n}$.
    This exercise is recommended for all readers.

### Problem 10

* 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?
Solutions