Linear Transformation

A mapping between 2 vector spaces that preserves vector addition and scalar multiplication.

Let $V,W$ be vector spaces over $F$. A linear transformation $T$ from $V$ to $W$ is such that:

• $T:V\rightarrow W$ be a function
• $T(u+v) = T(u) + T(v) \quad \forall u,v \in V$
• $T(au) = aT(u) \quad \forall u,v \in V$

Kernel

Denoted as $\text{ker}\;T$.

A subspace of $V$.

Null

Denoted by $\text{null}\;T$. Dimension of $\text{ker}\;T$.

Range

Denoted as $\text{ran}\;T$.

A subspace of $W$.

If $B$ is a basis of $V$, $\text{ran}\;T = \text{span}\;T(B)$.

Rank

Denoted by $\text{rank}\;T$. Dimension of $\text{ran}\;T$.

Rank-Nullity Theorem

Matrix of a Linear Transformation

Any linear transformation can be expressed as a matrix. And vice versa.

Suppose $V$ and $W$ are 2 vector spaces over $F$ with $m$ and $n$ dimensions respectively. And suppose a linear transformation $T: V \rightarrow W$ is defined on them. And suppose $(v_i)_{i=1}^m$ and $(w_j)_{j=1}^{n}$ be ordered bases of each vector spaces.

Matrix of $T$, is $A_{n\times m}$ ($\in F^{n\times m}$) such that: $\big(T(v_i)\big)_{i=1}^m = A_{m\times n} (w_j)_{j=1}^n$.

Change of Basis

Suppose now 2 different bases of $V$ and $W$ are given: $(v_i')_{i=1}^m$ and $(w_j')_{j=1}^{n}$. The new corresponding matrix $A'$ can be calculated using:

Here:

• $P_{n\times n}$ is the change-of-basis matrix that satisfies $(v_i)_{i=1}^n$
• $Q_{m\times m}$ is the change-of-basis matrix that satisfies $(w_j)_{j=1}^m$