線性代數 - 線性變換
Linear transformations¶
Definition:
Let \(V,W\) be two vector spaces and \(T\) be a map from \(V\) to \(W\)
\(T:V\to W\)
If \(T\) satisfies the following properties:
(a) \(T(v_1)+T(v_2)=T(v_1+v_2),\forall v_1,v_2\in V\)
(b) \(cT(v)=T(cv),\forall v\in V,c\in F\)
Then we call \(T\) a linear transformation.
Or we say \(T\) is linear in short.
In particular, we call the linear transformation \(T:V\to V\)"A linear operator on \(V\)".
Theorem:
Let \(T:V\to W\) be a linear transformation from \(V\) to \(W\)
then \(T(\mathit0_V)=\mathit0_W\), where \(\mathit0_V,\mathit0_W\) denotes the zero vector in \(V,W\), respectively.
Proof:
\(T(\mathit0_V)=T(\mathit0_V+\mathit0_V)=T(\mathit0_V)+T(\mathit0_V)\)
\(\therefore\mathit0_W=T(\mathit0_V)\)
Null space (Kernel) and Range of a linear map¶
Definition:
Let \(T:V\to W\) be a linear transformation.
The Null space of \(T\), denoted as \(N(T)\), is consisted with those vectors in \(V\) which has image \(\mathit0_W\) after applying the linear transformation.
Some uses the notation \(\text{ker}(T)\)
The Range of \(T\), denoted as \(R(T)\), is consisted with all the vectors \(w\) in \(W\) where there exsits some \(v\in V\) such that \(T(v)=w\).
Theorem:
Let \(T:V\to W\) be a linear transformation
\(N(T)\) is a subspace of \(V\) and \(R(T)\) is a subspace of \(W\).
Proof
Let \(u,v\in N(T),a\in F\)
\(\because T(au+v)=aT(u)+T(v)=a\mathit0_W+\mathit0_W=\mathit0_W\)
\(\therefore au+v\in N(T)\)
Therefore \(N(T)\) is a subspace of \(V\)
Let \(x,y\in R(T),b\in F\)
\(\because T(bx+y)=bT(x)+T(y)\) and \(bT(x)\in W,T(y)\in W\)
Since \(W\) is a vector space, \(bT(x)+T(y)\in W\)
\(\therefore bx+y\in R(T)\)
Therefore \(R(T)\) is a subspace of \(W\)
Definition:
Let \(V\) be a vector space and \(W\) be a subspace of it, and \(T:V\to V\) be linear.
By \(W\) is \(T\)-invariant, we mean that \(T(W)\subseteq W\).
Theorem:
Let \(V\) be a vector space and \(W\) be a subspace of it, and \(T:V\to V\) be linear.
Then the subsapces \(\{\mathit0\},N(T),R(T)\) are \(T\)-invariant.
Theorem:
Let \(T:V\to W\) be a linear transformation
If \(\beta=\{v_1,v_2,\dots v_n\}\) is a basis for \(V\), then \(R(T)=\textrm{span}(T(\beta))\).
This means that to understand the range of \(T\), we can observe what \(T\) would do to the basis of \(V\).
This is because every vector \(v\in V\) has a unique linear combination of the basis and most importantly, \(T\) is linear.
Proof
(\(\subseteq\))
Let \(w\in R(T)\)
\(w=T(v),v\in V\)
Let \(v=\displaystyle\sum_{k=1}^{n}a_kv_k\)
then \(w=T(\displaystyle\sum_{k=1}^{n}a_kv_k)=\displaystyle\sum_{k=1}^{n}a_kT(v_k)\in\textrm{span}(\{T(v_1),T(v_2),\dots,T(v_n)\})\)
(\(\supseteq\))
Let \(w\in\textrm{span}(T(\beta))\)
then \(w=\displaystyle\sum_{k=1}^{n}a_kT(v_k)=T(\displaystyle\sum_{k=1}^{n}a_kv_k)\)
Since \(\displaystyle\sum_{k=1}^{n}a_kv_k\in\textrm{span}(\{v_1,v_2,\dots,v_n\})=V\), therefore \(w\in R(T)\)
Therefore if \(\beta=\{v_1,v_2,\dots v_n\}\) is a basis for \(V\), then \(R(T)=\textrm{span}(T(\beta))\).
Definition:
Let \(T:V\to W\) be linear.
Define \(\text{nullity}(T)=\dim({N(T)})\) as the nullity of \(T\)
Define \(\text{rank}(T)=\dim(R(T))\) as the rank of \(T\)
The Dimension theorem¶
Let \(V,W\) be finite-dimensional vector spaces
Let \(T:V\to W\) be linear, then
\(\text{nullity}(T)+\text{rank}(T)=\dim{V}\)
Theorem:
Let \(V,W\) be finite-dimensional vector spaces
Let \(T:V\to W\) be linear
\(T\) is one-to-one if and only if \(N(T)=\{\mathit0_V\}\)
Proof
(\(\implies\))
Let \(x\in V,T(x)=\mathit0_W\)
Since \(T(\mathit0_V)=\mathit0_W\) and \(T\) is one-to-one, \(0_V=x\)
Therefore \(N(T)=\{\mathit0_V\}\)
(\(\Longleftarrow\))
Let \(x,y\in V,T(x)=T(y)\)
\(T(x)-T(y)=T(x-y)=\mathit0_W\implies x-y=\mathit0_V\)
Therefore \(T\) is one-to-one.
Let \(V,W\) be finite-dimensional vector spaces of equal dimensions.
Let \(T:V\to W\) be linear.
The following are equivalent.
(a) \(T\) is one-to-one
(b) \(T\) is onto
(c) \(\text{rank}(T)=\dim{V}\)
Since \(T\) is one-to-one, \(N(T)=\{\mathit0_V\}\)
\(\iff\text{rank}(T)=\dim{V}=\dim{W}
\\\iff R(T)=W\)
Matrix representation of linear transformation¶
Definition:
Let \(V\) be a fintie-dimensional vector space.
An ordered basis of \(V\) is a basis of \(V\) that has a specific order.
We can write \(\beta=(v_1,v_2,\dots,v_n)\) (n-tuple) and \(\text{span}(\beta):=\{\displaystyle\sum_{k=1}^{n}a_kv_k\mid a_k\in F,1\le k\le n\}=V\) and \(\beta\) is linearly independent.
Definition:
Let \(V\) be a fintie-dimensional vector space, and \(\beta=(v_1,v_2,\dots,v_n)\) be an ordered basis of \(V\).
For \(x\in V\), suppose \(x=\displaystyle\sum_{k=1}^{n}a_kv_k\) where \(a_k\) are the unique scalars for \(x\).
Define the coordinate vector of \(x\) relative to \(\beta\), denoted \([x]_{\beta}\) by \([x]_{\beta}=\begin{pmatrix}a_1\\a_2\\\vdots\\a_n\end{pmatrix}\)
Definition:
Let \(T:V\to W\) and \(\beta,\gamma\) be the ordered basis of \(V,W\), respectively.
We call the \(m\times n\) matrix \(A\), defined by \(A_{ij}=a_{ij}\), the matrix representation of \(T\) in the ordered basis \(\beta\) and \(\gamma\), and write \(A=[T]_{\beta}^{\gamma}\).
If \(V=W\) and \(\beta=\gamma\), we write \(A=[T]_{\beta}\) for convenience.
Definition:
The \(n\times n\) identity matrix is defined by \((I_n)_{ij}:=\delta_{ij}\), where \(\delta_{ij}=\begin{cases}1,&\text{if}\ i=j\\0,&\text{if}\ i\not=j\end{cases}\).
Definition:
Let \(T,U:V\to W\) be linear transformations, \(\forall v\in V,a\in F\), define
\((T+U)(v)=T(v)+U(v)\)
\((aT)(v)=a(T(v))\)
Definition:
\(\mathcal{L}(V,W)\) is the collection of all linear maps from \(V\) to \(W\), by the addidtion and scalar multiplication defined above, \(\mathcal{L}(V,W)\) is a vector space over \(F\).
When \(W=V\), we write \(\mathcal{L}(V)\) instead of \(\mathcal{L}(V,V)\)
For a vector \(x\in V\)
\(\hat{x}:V^*\to F\)
\(\hat{x}(f)=f(x),\forall f\in V^*\)
If \(\hat{x}(f)=0,\forall f\in V^*\) , then \(x=\mathit0\)