Linear transformations
Definition
Definition: let \(V\) and \(W\) be vector spaces, a mapping \(L: V \to W\) is a linear transformation or linear map if
\[ L(\lambda \mathbf{v}_1 + \mu \mathbf{v}_2) = \lambda L(\mathbf{v}_1) + \mu L(\mathbf{v}_2), \]for all \(\mathbf{v}_{1,2} \in V\) and \(\lambda, \mu \in \mathbb{K}\).
A linear transformation may also be called a vector space homomorphism. If the linear transformation is a bijection then it may be called a linear isomorphism.
In the case that the vector spaces \(V\) and \(W\) are the same; \(V=W\), a linear transformation \(L: V \to V\) will be referred to as a linear operator on \(V\) or linear endomorphism .
The image and kernel
Let \(L: V \to W\) be a linear transformation from a vector space \(V\) to a vector space \(W\). In this section the effect is considered that \(L\) has on subspaces of \(V\). Of particular importance is the set of vectors in \(V\) that get mapped into the zero vector of \(W\).
Definition: let \(L: V \to W\) be a linear transformation. The kernel of \(L\), denoted by \(\ker(L)\), is defined by
\[ \ker(L) = \{\mathbf{v} \in V \;|\; L(\mathbf{v}) = \mathbf{0}\}. \]
The kernel is therefore a set consisting of vectors in \(V\) that get mapped into the zero vector of \(W\).
Definition: let \(L: V \to W\) be a linear transformation and let \(S\) be a subspace of \(V\). The image of \(S\), denoted by \(L(S)\), is defined by
\[ L(S) = \{\mathbf{w} \in W \;|\; \mathbf{w} = L(\mathbf{v}) \text{ for } \mathbf{v} \in S \}. \]The image of the entire vector space \(L(V)\), is called the range of \(L\).
With these definitions the following theorem may be posed.
Theorem: if \(L: V \to W\) is a linear transformation and \(S\) is a subspace of \(V\), then
- \(\ker(L)\) is a subspace of \(V\).
- \(L(S)\) is a subspace of \(W\).
Proof:
Let \(L: V \to W\) be a linear transformation and \(S\) is a subspace of \(V\).
To prove 1, let \(\mathbf{v}_{1,2} \in \ker(L)\) and let \(\lambda, \mu \in \mathbb{K}\). Then
therefore \(\lambda \mathbf{v}_1 + \mu \mathbf{v}_2 \in \ker(L)\) and hence \(\ker(L)\) is a subspace of \(V\).
To prove 2, let \(\mathbf{w}_{1,2} \in L(S)\) then there exist \(\mathbf{v}_{1,2} \in S\) such that \(\mathbf{w}_{1,2} = L(\mathbf{v}_{1,2})\) For any \(\lambda, \mu \in \mathbb{K}\) we have
since \(\lambda \mathbf{v}_1 + \mu \mathbf{v}_2 \in S\) it follows that \(\lambda \mathbf{w}_1 + \mu \mathbf{w}_2 \in L(S)\) and hence \(L(S)\) is a subspace of \(W\).
Matrix representations
Theorem: let \(L: \mathbb{R}^n \to \mathbb{R}^m\) be a linear transformation, then there is an \(m \times n\) matrix \(A\) such that
\[ L(\mathbf{x}) = A \mathbf{x}, \]for all \(x \in \mathbb{R}^n\). With the \(i\)th column vector of \(A\) given by
\[ \mathbf{a}_i = L(\mathbf{e}_i), \]for a basis \(\{\mathbf{e}_1, \dots, \mathbf{e}_n\} \subset \mathbb{R}^n\) and \(i \in \{1, \dots, n\}\).
Proof:
For \(i \in \{1, \dots, n\}\), define
and let
If \(\mathbf{x} = x_1 \mathbf{e}_1 + \dots + x_n \mathbf{e}_n\) is an arbitrary element of \(\mathbb{R}^n\), then
It has therefore been established that each linear transformation from \(\mathbb{R}^n\) to \(\mathbb{R}^m\) can be represented in terms of an \(m \times n\) matrix.
Theorem: let \(E = \{\mathbf{e}_1, \dots, \mathbf{e}_n\}\) and \(F = \{\mathbf{f}_1, \dots, \mathbf{f}_n\}\) be two ordered bases for a vector space \(V\), and let \(L: V \to V\) be a linear operator on \(V\), \(\dim V = n \in \mathbb{N}\). Let \(S\) be the \(n \times n\) transition matrix representing the change from \(F\) to \(E\),
\[ \mathbf{e}_i = S \mathbf{f}_i, \]for \(i \in \mathbb{N}; i\leq n\).
If \(A\) is the matrix representing \(L\) with respect to \(E\), and \(B\) is the matrix representing \(L\) with respect to \(F\), then
\[ B = S^{-1} A S. \]
Proof:
Will be added later.
Definition: let \(A\) and \(B\) be \(n \times n\) matrices. \(B\) is said to be similar to \(A\) if there exists a nonsingular matrix \(S\) such that \(B = S^{-1} A S\).
It follows from the above theorem that if \(A\) and \(B\) are \(n \times n\) matrices representing the same operator \(L\), then \(A\) and \(B\) are similar.