# Lecture - Linear Algebra MT22, XI

> Source: https://ollybritton.com/notes/uni/prelims/mt22/linear-algebra/lectures/11/ · Updated: 2025-01-05 · Tags: uni, lecture

### Flashcards
What is true about the invertibility of any change of basis matrix?::
Any change of basis matrix is invertible.

Are transformation matrices different with respect to different bases?::
Yes.

If you have a transformation matrix $B$ representing the transformation $T$ with respect to a basis $\mathcal{E}$, and you have change of basis matrix $X$ that converts from $\mathcal{E}$ to $\mathcal{F}$, what would be the transformation matrix for $T$ with respect to $F$?::
$$
X^{-1}BX
$$

What is the definition of $A, B$ being similar matrices?::
There exists an invertible matrix $X$ such that $A = X^{-1}BX$.

What do similar (in the technical sense) matrices represent?::
The same linear transformation with respect to different bases.

What is true about the relation $A \sim B$ where $A \sim B$ if $A, B$ are similar matrices?::
It is an equivalance relation.

What is true about invertiblitity for two similar matrices $A \sim B$?::
$$
A^{-1} \text{ exists} \iff B^{-1} \text{ exists}
$$

If $U, V, W$ are finite dimensional vector spaces over the same field with ordered bases $\mathcal{B, E, F}$ and $S : U \to V$, $T : V \to W$ are linear maps, with a matrix $A$ representing $S$ from $U$ w.r.t. $\mathcal{B}$ to $V$ and $B$ representing $T$ from $V$ w.r.t. $E$ to $W$ w.r.t. $\mathcal{F}$, then what does $C = BA$ represent?::
$T \cdot S$ from $U$ w.r.t. $\mathcal{B}$ to $W$ w.r.t. $\mathcal{F}$.

Why is matrix multiplication “natural”?::
It represents the composition of linear maps.

---
Olly Britton — https://ollybritton.com. Machine-readable index: https://ollybritton.com/llms.txt