Linear Algebra I MT22, Change of basis theorem
Composition of linear maps as matrix multiplication
If
\[T : V \to W\]
What does the notation
\[A = {} _ \mathcal{U}T _ \mathcal{V}\]
mean?
The transformation matrix of $T$ with respect to basis $\mathcal{V}$ for $V$ and $\mathcal{U}$ for $W$.
There is a theorem that states
Let $U, V, W$ be finite-dimensional vector spaces with dimensions $m, n, p$ and ordered bases $\mathcal{U}, \mathcal{V}, \mathcal{W}$. Let $S : U \to V$ and $T : V \to W$ be linear.
Let $A = {} _ \mathcal{V}S _ \mathcal{U}$ and $B = {} _ \mathcal{W}T _ \mathcal{V}$. Then $BA = {} _ \mathcal{W}TS _ \mathcal{U}$.
What does this actually mean?
Matrix multiplication represents the composition of linear maps.
Let $V$ be an $n$-dimensional vector space with ordered basis $\mathcal{V} = [v _ 1, \ldots, v _ n]$.
Let $W$ be an $m$-dimensional vector space with ordered basis $\mathcal{W} = [w _ 1, \ldots, w _ m]$.
Let $T : V\to W$ be a linear transformation. Let ${} _ \mathcal{W}T _ \mathcal{V}$ be the matrix representing this linear transformation.
Say $v \in V$ has coordinate representation $\underline{x}$.
What does ${} _ \mathcal{W}T _ \mathcal{V} \underline{x}$ equal?
$\underline{y}$ where $\underline{y}$ is the coordinate representation of $T(v)$.
Let $V$ be an $n$-dimensional vector space with ordered basis $\mathcal{V} = [v _ 1, \ldots, v _ n]$.
Let $W$ be an $m$-dimensional vector space with ordered basis $\mathcal{W} = [w _ 1, \ldots, w _ m]$.
Let $T : V\to W$ be a linear transformation. Let ${} _ \mathcal{W}T _ \mathcal{V}$ be the matrix representing this linear transformation. Denote the entries of ${} _ \mathcal{W}T _ \mathcal{V}$ with $(a _ {ij})$.
What’s the maximum value $i$ could be?
Let $V$ be an $n$-dimensional vector space with ordered basis $\mathcal{V} = [v _ 1, \ldots, v _ n]$.
Let $W$ be an $m$-dimensional vector space with ordered basis $\mathcal{W} = [w _ 1, \ldots, w _ m]$.
Let $T : V\to W$ be a linear transformation. Let ${} _ \mathcal{W}T _ \mathcal{V}$ be the matrix representing this linear transformation. Denote the entries of ${} _ \mathcal{W}T _ \mathcal{V}$ with $(a _ {ij})$.
What’s the maximum value $j$ could be?
Let $V$ be an $n$-dimensional vector space with ordered basis $\mathcal{V} = [v _ 1, \ldots, v _ n]$.
Let $W$ be an $m$-dimensional vector space with ordered basis $\mathcal{W} = [w _ 1, \ldots, w _ m]$.
Let $T : V\to W$ be a linear transformation. Let ${} _ \mathcal{W}T _ \mathcal{V}$ be the matrix representing this linear transformation. Denote the entries of ${} _ \mathcal{W}T _ \mathcal{V}$ with $(a _ {ij})$.
What is in the $i$-th column of the matrix?
$\underline{y _ i}$ where $\underline{y _ i}$ is the coordinate representation of $T(v _ i)$.
Let $V$ be an $n$-dimensional vector space with ordered basis $\mathcal{V} = [v _ 1, \ldots, v _ n]$.
Let $W$ be an $m$-dimensional vector space with ordered basis $\mathcal{W} = [w _ 1, \ldots, w _ m]$.
Let $T : V\to W$ be a linear transformation. Let ${} _ \mathcal{W}T _ \mathcal{V}$ be the matrix representing this linear transformation. Denote the entries of ${} _ \mathcal{W}T _ \mathcal{V}$ with $(a _ {ij})$.
What property characterises $T(v _ i)$?
Let $V$ be an $n$-dimensional vector space with ordered basis $\mathcal{V} = [v _ 1, \ldots, v _ n]$.
Let $W$ be an $m$-dimensional vector space with ordered basis $\mathcal{W} = [w _ 1, \ldots, w _ m]$.
Let $T : V\to W$ be a linear transformation. Let ${} _ \mathcal{W}T _ \mathcal{V}$ be the matrix representing this linear transformation. Denote the entries of ${} _ \mathcal{W}T _ \mathcal{V}$ with $(a _ {ij})$.
How can you remember the property characterises $T(v _ i)$, namely
\[T(v _ i) = \sum^m _ {k=1} a _ {ki} w _ k\]
?
This is just a mathematical way of stating the fact that the $i$-th column is the coordinate representation of $T(v _ i)$.
Let $U, V, W$ be finite-dimensional vector spaces with dimensions $m, n, p$ and ordered bases $\mathcal{U}, \mathcal{V}, \mathcal{W}$. Let $S : U \to V$ and $T : V \to W$ be linear.
Let $A = {} _ \mathcal{V}S _ \mathcal{U}$ and $B = {} _ \mathcal{W}T _ \mathcal{V}$. Then what is true?
Then $BA = {} _ \mathcal{W}TS _ \mathcal{U}$.
Let $U, V, W$ be finite-dimensional vector spaces with dimensions $m, n, p$ and ordered bases $\mathcal{U}, \mathcal{V}, \mathcal{W}$. Let $S : U \to V$ and $T : V \to W$ be linear. What transformation would the matrix ${} _ \mathcal{W}TS _ \mathcal{U}$ represent?
When proving the following
Let $U, V, W$ be finite-dimensional vector spaces with dimensions $m, n, p$ and ordered bases $\mathcal{U}, \mathcal{V}, \mathcal{W}$. Let $S : U \to V$ and $T : V \to W$ be linear.
Let $A = {} _ \mathcal{V}S _ \mathcal{U}$ and $B = {} _ \mathcal{W}T _ \mathcal{V}$. Then $BA = {} _ \mathcal{W}TS _ \mathcal{U}$.
You start by writing down the property that characterises the matrices $A$ and $B$, namely
\[S(u _ i) = \sum^n _ {k=1} a _ {ki} v _ k\]
and
\[T(v _ i) = \sum _ {k=1}^p b _ {ki} w _ k\]
What do you need to end up showing?
There’s a very important result about matrices and the composition of linear maps that’s useful for proving the change of basis theorem. @State it in full.
Let $U, V, W$ be finite-dimensional vector spaces with dimensions $m, n, p$ and ordered bases $\mathcal{U}, \mathcal{V}, \mathcal{W}$. Let $S : U \to V$ and $T : V \to W$ be linear.
Let $A = {} _ \mathcal{V}S _ \mathcal{U}$ and $B = {} _ \mathcal{W}T _ \mathcal{V}$. Then $BA = {} _ \mathcal{W}TS _ \mathcal{U}$.
Change of basis theorem
@State the change of basis theorem in terms of $V, \mathcal{V}, \mathcal{V}', W, \mathcal{W}, \mathcal{W}'$ and $T$.
Let $V$ be a finite-dimensional vector space with ordered bases $\mathcal{V}, \mathcal{V}'$. Let $W$ be a finite-dimensional vector space with ordered bases $\mathcal{W}, \mathcal{W}'$. Let $T : V \to W$ be a linear map. Then
\[{} _ \mathcal{W'}T _ \mathcal{V'} = ({} _ \mathcal{W'}I _ \mathcal{W})({} _ \mathcal{W}T _ \mathcal{V})({} _ \mathcal{V}I _ \mathcal{V'})\]For a matrix like ${} _ \mathcal{V}T _ \mathcal{V'}$, what’s the basis of the input?
Bite-sized
For an endomorphism $T : V \to V$ and two bases $\mathcal V, \mathcal V'$, the change-of-basis formula gives a similarity transformation: $[T] _ {\mathcal V'} = P^{-1} [T] _ {\mathcal V} P$, where $P$ is the change-of-basis matrix from $\mathcal V'$ to $\mathcal V$.
Why is similarity the right notion of equivalence between matrices in NLA?
Similar matrices represent the same linear transformation in different bases. So invariants of the linear map — eigenvalues, determinant, trace, characteristic polynomial, minimal polynomial — are preserved. Algorithms like the QR algorithm work by applying a sequence of similarity transformations (specifically orthogonal ones, for stability) to reveal the eigenvalues on the diagonal of the limit. The whole structure-revealing program of NLA is “find an orthogonal similarity that diagonalises (or triangularises) $A$”.