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.

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.

1. Prove that matrix addition represents addition of linear maps.
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 $c_{1}t_{1}+\dots +c_{6}t_{6}$ 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.

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