Lecture - Linear Algebra I MT22, XI
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 $F$?
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 $A \sim B$?
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.