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.