Module III·Article II·~1 min read
Linear Transformations and Matrices
Vector Spaces
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Linear Transformations
A mapping $f: V \to W$ between vector spaces is called linear (a homomorphism) if:
- $f(u + v) = f(u) + f(v)$
- $f(\alpha v) = \alpha f(v)$
Linear transformations are “structure-preserving” mappings of vector spaces.
Kernel and Image
Kernel: $\ker f = {v \in V : f(v) = 0}$ — a subspace of $V$.
Image: $\mathrm{Im}, f = {f(v) : v \in V}$ — a subspace of $W$.
Rank–nullity theorem: $\dim V = \dim(\ker f) + \dim(\mathrm{Im}, f)$.
This is the “dimension conservation law”: what is “lost” ($\ker f$) plus what is “achieved” ($\mathrm{Im}, f$) equals the original.
Matrix of a Linear Transformation
Fix bases $B$ in $V$ and $C$ in $W$. The matrix $A$ of a linear transformation $f$ is the matrix whose columns are the coordinates of $f(b_1), \ldots, f(b_n)$ in the basis $C$.
Then $[f(v)]_C = A \cdot [v]_B$.
Every linear transformation between finite-dimensional spaces is specified by a matrix (with fixed bases), and vice versa.
Change of Basis
If $B$ and $B'$ are two bases of $V$, the change of basis matrix $S$ converts coordinates from $B$ to $B'$. The matrix of the transformation in the new basis is: $A' = S^{-1}AS$.
Two operators $A$ and $A'$ are similar if $A' = S^{-1}AS$. Similar matrices represent the same operator in different bases.
Injection, Surjection, Isomorphism
$f$ is an injection $\iff \ker f = {0} \iff \dim(\ker f) = 0 \iff \operatorname{rank} A = n$.
$f$ is a surjection $\iff \mathrm{Im}, f = W \iff \dim(\mathrm{Im}, f) = \dim W \iff \operatorname{rank} A = \dim W$.
$f$ is an isomorphism $\iff$ bijection $\iff \dim V = \dim W$ and $A$ is a nondegenerate (invertible) matrix.
All $n$-dimensional spaces over $F$ are isomorphic to $F^n$.
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