Notes - Linear Algebra I MT22, Change of basis theorem

Created: January 07, 2023

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 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 entires of ${} _ \mathcal{W}T _ \mathcal{V}$ with $(a _ {ij})$.

What’s the maximum value $i$ could be?
\[m\]

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 entires of ${} _ \mathcal{W}T _ \mathcal{V}$ with $(a _ {ij})$.

What’s the maximum value $j$ could be?
\[n\]

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 entires 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 entires of ${} _ \mathcal{W}T _ \mathcal{V}$ with $(a _ {ij})$.

What property characterises $T(v _ i)$?
\[T(v_i) = \sum^m_{k=1} a_{ki} w_k\]

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 entires 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?
\[(T \circ S)\]

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

\[T(v_i) = \sum_{k=1}^n b_{ki} w_k\]

What do you need to end up showing?
\[\begin{aligned} (T \circ S)(u_i) &= T(S(u_i)) \\\\ &= \ldots \\\\ &= \sum_{k=1}^p (BA)_{ki}w_k \end{aligned}\]

There’s a very important result about matrices and the composition of linear maps that’s useful for proving the change of basis theorem. Can you 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

Can you 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?
\[\mathcal{V}’\]


Related posts