Linear Transformations

Transformations

A transformation $T$ from a set $X$ (domain) to a set $Y$ (codomain) is a rule that for each argument $x\in X$ assigns a value $y\in Y$.
We write $T:X\to Y$ to say that $T$ is a transformation with domain $X$ and codomain $Y$.

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Linear transformations

Let $V,W$ be vector spaces over the same field $\mathbb{F}$. A transformation $T:V\to W$ is called linear if

1. $T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})\forall \mathbf{u},\mathbf{v}\in V$
2. $T(\alpha \mathbf{v})=\alpha T(\mathbf{v})$

Properties 1 and 2 together are equivalent to $T(\alpha \mathbf{u}+\beta \mathbf{v})=\alpha T(\mathbf{u})+\beta T(\mathbf{v})\forall \mathbf{u},\mathbf{v}\in V\text{ and }\alpha ,\beta \in \mathbb{F}$.
In other words, $T$ applied on a linear combination of $\mathbf{u}$ and $\mathbf{v}$ should equal the same linear combination of the transformed versions of $\mathbf{u}$ and $\mathbf{v}$.

Properties of linear transformations

• $T(0)=0$.
• $T(V)$ is a vector space (subspace of $W$).
• $\text{Ker }T$ is a subspace of $V$.

Matrix-vector multiplication as a linear transformation

A linear transformation $T:{\mathbb{F}}^n\to {\mathbb{F}}^m$ can be represented as “multiplication” by a matrix.

Claim: It is sufficient to know how $T$ acts on the standard basis ${\mathbf{e}}_1,{\mathbf{e}}_2,\dots {\mathbf{e}}_n$ of ${\mathbb{F}}^n$. Namely, it is sufficient to know $n$ vectors in ${\mathbb{F}}^m$,

$${\mathbf{a}}_1:=T({\mathbf{e}}_1),{\mathbf{a}}_2:=T({\mathbf{e}}_2),\dots ,{\mathbf{a}}_n:=T({\mathbf{e}}_n).$$

This is easy to show. Let $\mathbf{x}=(x_1,x_2,\dots ,x_n{)}^T$ . Then, $\mathbf{x}={\sum }_{k=1}^n{x}_k{\mathbf{e}}_k$, and

$$T(\mathbf{x})=T(\sum _{k=1}^n{x}_k{\mathbf{e}}_k)=\sum _{k=1}^{n}T(x_k{\mathbf{e}}_k)=\sum _{k=1}^n{x}_{k}T({\mathbf{e}}_k)=\sum _{k=1}^n{x}_k{\mathbf{a}}_k.$$

So, if we join the vectors (columns) ${\mathbf{a}}_1,{\mathbf{a}}_2,\dots ,{\mathbf{a}}_n$ together in a matrix $A=[{\mathbf{a}}_1,{\mathbf{a}}_2,\dots ,{\mathbf{a}}_n]$, this matrix contains all the information about $T$. We package $a_i$ together in manner because it makes it easy to compute $T(\mathbf{x})$ when we define the product of a matrix and a vector like so:

$$A\mathbf{x}=\sum _{k=1}^n{x}_k{\mathbf{a}}_k=x_1\left(\begin{array}{c}{a}_{1,1}\\ a_{2,1}\\ ⋮\\ a_{m,1}\end{array}\right)+x_2\left(\begin{array}{c}{a}_{1,2}\\ a_{2,2}\\ ⋮\\ a_{m,2}\end{array}\right)+\dots +x_n\left(\begin{array}{c}{a}_{1,n}\\ a_{2,n}\\ ⋮\\ a_{m,n}\end{array}\right)$$

So, by this column by coordinate definition of matrix by vector multiplication, we have $T(\mathbf{x})=A\mathbf{x}$, where the columns of $A$ are the standard basis vectors of ${\mathbb{F}}^n$ transformed by $T$.

Things

• A linear transformation $T$ is completely defined by its values on the standard basis in its domain. In fact, the basis need not be standard - one can consider any basis, even a generating set (every generating set has a basis, right?)
• Matrix multiplication did not just “happen” to be a linear transformation; it was defined specifically to be one.
• It naturally arises from this definition that the number of columns of the matrix must equal the number of entries in the vector.
• If $T$ is a linear transformation from ${\mathbb{F}}^n\to {\mathbb{F}}^r$, $T$ is represented by a $r\times n$ matrix

Note

Often no distinction is made between a linear transformation and its matrix. $T\mathbf{x}$ may mean applying $T$ on $\mathbf{x}$ or multiplying $\mathbf{x}$ by the matrix of $T$.

Important

We previously claimed that it is sufficient to know how a linear map $T$ acts on a basis of its domain to know how it acts on any member of its domain. Taking this further, it is also true that if we merely assign an element from a vector space $B$ to every element of a basis of a vector space $A$, we get a linear map $L:A\to B$.

Linear transformations as a vector space

If we fix vector spaces $V$ and $W$ and consider the collection of all linear transformations from $V$ to $W$ (denoted by $\mathcal{L}(V,W)$), we can define two operations on $\mathcal{L}(V,W)$: multiplication by a scalar and addition like so

1. $(\alpha T)\mathbf{v}=\alpha (T\mathbf{v})\forall \mathbf{v}\in V\forall T\in \mathcal{L}(V,W)$
2. $(T_1+T_2)\mathbf{v}=T_1\mathbf{v}+T_2\mathbf{v}\forall \mathbf{v}\in V\forall T\in \mathcal{L}(V,W)$

It can be easily shown that these operations satisfy the axioms of a vector space. Thus, $\mathcal{L}({\mathbb{F}}^n,{\mathbb{F}}^m)$ is a vector space.

While we usually do not define a product for vector spaces, a useful definition of multiplication does exist for some vector spaces, like this one.

Definition 1(Definition).

If $T\in \mathcal{L}(U,V)$ and $S\in \mathcal{L}(V,W)$ then the product $ST\in \mathcal{L}(U,W)$ is defined by

$$(ST)(\mathbf{u})=S(T\mathbf{u})$$

for $\mathbf{u}\in U$.

$ST$ is just the usual composition $S\circ T$ of two functions.

When the product is defined, the following properties can be easily proved:

• Associativity: $T_1(T_2{T}_3)=(T_1{T}_2)T_3$
• Identity: $TI=IT=T$
• distributivity: $(S_1+S_2)T=S_{1}T+S_{2}T$, $T(S_1+S_2)=T{S}_1+T{S}_2$.

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Matrix multiplication

Knowing matrix-vector multiplication, a natural way arises to define the product $AB$ of two matrices: multiply by $A$ each column of $B$, and join the resulting column vectors into a matrix.
If ${\mathbf{b}}_1,{\mathbf{b}}_2,\dots ,{\mathbf{b}}_r$ are the columns of $B$, then $A{\mathbf{b}}_1,A{\mathbf{b}}_2,\dots ,A{\mathbf{b}}_r$ are the columns of $AB$.
It is clear that in order for the multiplication to be defined, the number of columns in $A$ must equal the number of rows in $B$ (LADW, p20 for a better explanation of why this is natural).

Matrix multiplication is associative and distributive. One can also take scalar multiples out. In general, it is not commutative.