Linear maps

COMP1021 MCS: Linear algebra

Previously

See the concept diagram

Things in pink we will look at today (and next time)

Questions

• From the practicals last week?
• From https://pollev.com/stevenaeola

Basis representation

• Remember: a set of vectors is a basis for the vector space if it spans the whole space and is linearly independent

• Thm For a vector space $V$ the representation of any $\mathbf{v} \in V$ as a linear combination of a given basis $S = \lbrace \mathbf{s}_1,\ldots,\mathbf{s}_n \rbrace$ is unique

• Proof (sketch). Suppose not. Then subtract the two representations. This violates one of the properties of a basis. So we have a contradiction.

Canonical (standard) basis

• This is kind of obvious for the canonical basis for $\Bbb{R}^n$ $$\lbrace \mathbf{e}_i: i \leq 1 \leq n \rbrace$$
• Where $\mathbf{e}_i$ is the vector with a 1 in the ith position and 0 elsewhere
• In 3D sometimes referred to as $\mathbf{e}_x,\mathbf{e}_y,\mathbf{e}_z$ or $\mathbf{i},\mathbf{j},\mathbf{k}$

Linear map

Defn If we have a function $f:V \rightarrow W$ where $V$ and $W$ are vectors spaces over a field $F$ then $f$ is a linear map if for any vectors $\mathbf{u}, \mathbf{v} \in V$ and $a \in F$

$$f(\mathbf{u} + \mathbf{v}) = f(\mathbf{u}) + f(\mathbf{v})$$

$$f(a \mathbf{v}) = a f(\mathbf{v}) $$

• This means that a linear map respects linear combinations
• Linear maps are sometimes called linear transformations or vector space homomorphisms
• If $V=W$ then $f$ is an endomorphism
• All linear maps preserve lines

Examples of linear maps

• Scaling (on $\Bbb{R}^3$)
• Reflection
• Rotation
• Shearing
• Projection
• Embedding
• Differentiation (on polynomials)

Linear maps and basis vectors

Every vector $\mathbf{v}$ can be represented as a linear combination of basis vectors $\mathbf{s}_i \in S$. If we apply a linear map $f$ then

$$f(\mathbf{v}) = f(\sum_{i=1}^n a_i \mathbf{s}_i)$$

$$ = \sum_{i=1}^n a_i f(\mathbf{s}_i)$$

So a linear map is characterised by its action on basis vectors

Images of basis vectors

For example, rotation by 90 degrees anti-clockwise in $\Bbb{R}^2$

• Q: What are the images of the canonical basis vectors $\mathbf{e}_1 (1,0)$ and $\mathbf{e}_2 (0,1)$?
• A: $(0,1)$ and $(-1,0)$ or $\mathbf{e}_2$ and $-1\mathbf{e}_1$
• Turn these into column vectors to get the matrix representation

$$\begin{pmatrix}0&-1 \\1 & 0 \end{pmatrix}$$

You try

Choose one of the linear map examples

Write down the images of the basis vectors in either $\Bbb{R}^2$ or $\Bbb{R}^3$

Calculating from basis vector images

What happens when we rotate the vector $(2,3)$?

$$\begin{aligned} rotate((2,3)) &= rotate(2\mathbf{e}_1 + 3\mathbf{e}_2) \\\ &= 2.rotate(\mathbf{e}_1) + 3.rotate(\mathbf{e}_2)\\\ &= 2\mathbf{e}_2 + 3.(-1.\mathbf{e}_1) \\\ &= (-3,2) \end{aligned}$$

Generalising basis images

Write basis vector images columns under f as a matrix

$$\small{ A = \begin{pmatrix} a_{11} & a_{12} & \ldots & a_{1n}\\a_{21} & a_{22} & \ldots & a_{2n}\\\ \vdots & \vdots & \ddots & \vdots \\a_{m1} & a_{m2} & \ldots & a_{mn} \end{pmatrix}}$$

$a_{ij}$ is ith component of f-image of jth basis vector

$$f(\mathbf{e_j}) = \sum_{i=1}^m a_{ij} \mathbf{e}_i$$

The image of the vector $A\mathbf{x} = A(x_1,\ldots,x_n)$ is

$$ f ( \sum_{j=1}^n x_j \mathbf{e}j) = (\sum{j=1}^n x_j f(\mathbf{e}_j))$$

$$= \sum_{j=1}^n x_j \sum_{i=1}^m a_{ij} \mathbf{e}_i$$

$$= \sum_{j=1}^n \sum_{i=1}^m a_{ij} x_j \mathbf{e}_i$$

So the ith component of the image of $\mathbf{x}$ is $$\sum_{j=1}^m a_{ij} x_j$$

Example

$$\begin{pmatrix} 1 & -2 & 3 \\ -4 & 5 & -6 \end{pmatrix} \begin{pmatrix} 9 \\ 8 \\7 \end{pmatrix} = \begin{pmatrix} 1 \times 9 -2 \times 8 + 3 \times 7 \\ -4 \times 9 +5 \times 8 -6 \times 7\end{pmatrix} = \begin{pmatrix} 14 \\ -38\end{pmatrix} $$

What are the dimensions of the domain and codomain?

Identity matrix

What is the matrix representation of the identity map?

All basis vectors map to themselves via the identity matrix $I$

$$\small{ I = \begin{pmatrix} 1 & 0 & \ldots & 0 \\ 0 & 1 & \ldots & 0\\\ \vdots & \vdots & \ddots & \vdots \\0 & 0 & \ldots & 1 \end{pmatrix}}$$

$I$ is a diagonal matrix: $i \neq j \implies a_{ij}=0$

Combining linear maps

• Suppose we have two linear maps $f$ and $g$ represented by matrices $A$ and $B$
• What is the matrix for $(f \circ g)(\mathbf{v}) = f(g(\mathbf{v}))$?
• Need to find images of basis vectors
• $B$ columns contains images of basis vectors under $g$
• So apply $A$ to columns of $B$ in turn and write as columns
• This gives the matrix of the map $f \circ g$ which we write $AB$

Matrix multiplication

Given matrices $A \in \Bbb{R}^{m \times n}$ and $B \in \Bbb{R}^{n \times k}$

The elements $c_{ij}$ of the product $C = AB \in \Bbb{R}^{m \times k}$ are

$$c_{ij} = \sum_{l=1}^{n} a_{il}b_{lj}$$

For $1 \leq i \leq m, 1 \leq j \leq k$

This also formalises our definition of multiplying a matrix by a vector $k=1$

Example

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

Check your answer example 2.3 from MML book

More

• 3B1B Chapter 3: Linear transformations and matrices
• 3B1B Chapter 4: Matrix multiplication as composition
• MML book section 2.2

Next time: Determinants and inverses

Practicals: Span, independence, basis, linear maps, matrices