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.

\overset{→}{v} \ \overset{f + g}{⟼} \ f ( \overset{→}{v} ) + g ( \overset{→}{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 : {ℝ}^{2} → {ℝ}^{3} are represented
with respect to the bases B and D by these matrices.

F = {R e p}_{B , D} ( f ) = {( \begin{matrix}1 & 3 \\ 2 & 0 \\ 1 & 0\end{matrix} )}_{B , D} \ G = {R e p}_{B , D} ( g ) = {( \begin{matrix}0 & 0 \\ − 1 & − 2 \\ 2 & 4\end{matrix} )}_{B , D}

Then, for any \overset{→}{v} ∈ V represented with respect to B,
computation of the representation of f ( \overset{→}{v} ) + g ( \overset{→}{v} )

( \begin{matrix}1 & 3 \\ 2 & 0 \\ 1 & 0\end{matrix} ) ( \begin{matrix}{v}_{1} \\ {v}_{2}\end{matrix} ) + ( \begin{matrix}0 & 0 \\ − 1 & − 2 \\ 2 & 4\end{matrix} ) ( \begin{matrix}{v}_{1} \\ {v}_{2}\end{matrix} ) = ( \begin{matrix}1 {v}_{1} + 3 {v}_{2} \\ 2 {v}_{1} + 0 {v}_{2} \\ 1 {v}_{1} + 0 {v}_{2}\end{matrix} ) + ( \begin{matrix}0 {v}_{1} + 0 {v}_{2} \\ − 1 {v}_{1} − 2 {v}_{2} \\ 2 {v}_{1} + 4 {v}_{2}\end{matrix} )

gives this representation of f + g \ ( \overset{→}{v} ).

( \begin{matrix}( 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{matrix} ) = ( \begin{matrix}1 {v}_{1} + 3 {v}_{2} \\ 1 {v}_{1} − 2 {v}_{2} \\ 3 {v}_{1} + 4 {v}_{2}\end{matrix} )

Thus, the action of f + g is described by this matrix-vector product.

{( \begin{matrix}1 & 3 \\ 1 & − 2 \\ 3 & 4\end{matrix} )}_{B , D} {( \begin{matrix}{v}_{1} \\ {v}_{2}\end{matrix} )}_{B} = {( \begin{matrix}1 {v}_{1} + 3 {v}_{2} \\ 1 {v}_{1} − 2 {v}_{2} \\ 3 {v}_{1} + 4 {v}_{2}\end{matrix} )}_{D}

This matrix is the entry-by-entry sum of original matrices, e.g., the {}1 , 1 entry of {R e p}_{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

{R e p}_{B , D} ( t ) = {( \begin{matrix}1 & 0 \\ 1 & 1\end{matrix} )}_{B , D} \ \text{so that} \ \overset{→}{v} = {( \begin{matrix}{v}_{1} \\ {v}_{2}\end{matrix} )}_{B} ↦ {( \begin{matrix}{v}_{1} \\ {v}_{1} + {v}_{2}\end{matrix} )}_{D} = t ( \overset{→}{v} )

then the scalar multiple map {}5 t acts in this way.

\overset{→}{v} = {( \begin{matrix}{v}_{1} \\ {v}_{2}\end{matrix} )}_{B} \ ⟼ \ {( \begin{matrix}5 {v}_{1} \\ 5 {v}_{1} + 5 {v}_{2}\end{matrix} )}_{D} = 5 ⋅ t ( \overset{→}{v} )

Therefore, this is the matrix representing {}5 t.

{R e p}_{B , D} ( 5 t ) = {( \begin{matrix}5 & 0 \\ 5 & 5\end{matrix} )}_{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 → 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 → W is represented with respect to B , D by H + G, and the map r ⋅ h : V → W is represented with respect to B , D by r H.

Proof

Problem 2; generalize the examples above.

A notable special case of scalar multiplication is multiplication by zero.
For any map {}0 ⋅ h is the zero homomorphism and for any matrix
{}0 ⋅ 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 \ × \ 3 zero matrix

Z = ( \begin{matrix}0 & 0 & 0 \\ 0 & 0 & 0\end{matrix} )

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{matrix}5 & − 1 & 2 \\ 6 & 1 & 1\end{matrix} ) + ( \begin{matrix}2 & 1 & 4 \\ 3 & 0 & 5\end{matrix} )
• {}6 ⋅ ( \begin{matrix}2 & − 1 & − 1 \\ 1 & 2 & 3\end{matrix} )
• ( \begin{matrix}2 & 1 \\ 0 & 3\end{matrix} ) + ( \begin{matrix}2 & 1 \\ 0 & 3\end{matrix} )
• {}4 ( \begin{matrix}1 & 2 \\ 3 & − 1\end{matrix} ) + 5 ( \begin{matrix}− 1 & 4 \\ − 2 & 1\end{matrix} )
• {}3 ( \begin{matrix}2 & 1 \\ 3 & 0\end{matrix} ) + 2 ( \begin{matrix}1 & 1 & 4 \\ 3 & 0 & 5\end{matrix} )

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 ⋅ G = Z
• ( r + s ) G = r G + s G
• Matrices have an additive inverse G + ( − 1 ) ⋅ G = Z.
• r ( G + H ) = r G + r H
• ( r s ) G = r ( s G )

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 ℒ ( V , W ) of linear maps from V to
W is isomorphic to {ℳ}_{m \ × \ 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} , … , {t}_{6} : {ℝ}^{2} → {ℝ}^{2}
there are scalars {c}_{1} , … , {c}_{6} ∈ ℝ such that
{c}_{1} {t}_{1} + … + {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 \text{trace} ( H + G ) = \text{trace} ( H ) + \text{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 )}^{t r a n s} = {G}^{t r a n s} + {H}^{t r a n s}
• {( r ⋅ H )}^{t r a n s} = r ⋅ {H}^{t r a n s}

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}^{t r a n s} is symmetric.
  Does every symmetric matrix have this form?
• Prove that the set of n \ × \ n symmetric matrices is

a subspace of {ℳ}_{n \ × \ 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

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