# Linear Algebra/Sums and Scalar Products

Linear Algebra
 ← Matrix Operations Sums and Scalar Products Matrix Multiplication →

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.

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

Prove Theorem 1.5.

2. 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.

1. Matrix addition is commutative $G+H=H+G$.
2. Matrix addition is associative $G+(H+J)=(G+H)+J$.
3. The zero matrix is an additive identity $G+Z=G$.
4. $0\cdot G=Z$
5. $(r+s)G=rG+sG$
6. Matrices have an additive inverse $G+(-1)\cdot G=Z$.
7. $r(G+H)=rG+rH$
8. $(rs)G=r(sG)$
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 $c_1t_1+\dots+c_6t_6$ 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.

1. ${{(G+H)}^{\rm trans}}={{G}^{\rm trans}}+{{H}^{\rm trans}}$
2. ${{(r\cdot H)}^{\rm trans}}=r\cdot{{H}^{\rm trans}}$
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.

1. Prove that for any $H$, the matrix $H+{{H}^{\rm trans}}$ is symmetric. Does every symmetric matrix have this form?
2. 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
1. How does matrix rank interact with scalar multiplication— can a scalar product of a rank $n$ matrix have rank less than $n$? Greater?
2. How does matrix rank interact with matrix addition— can a sum of rank $n$ matrices have rank less than $n$? Greater?

Solutions

Linear Algebra
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