# Lecture - Linear Algebra I MT22, X

### Flashcards

What is $\text{Ker}(T)$ for $T : \Pi _ n \to \Pi _ n$ given by $T(v) = v’$?

What is $\text{Im}(T)$ for $T : \Pi _ n \to \Pi _ n$ given by $T(v) = v’$?

For a projection $T : V\to V$ given by $T(v) = w$ when $v = u + w$ for a vector space where $V = U \oplus W$, what is $\text{Ker}(T)$?

For a projection $T : V\to V$ given by $T(v) = w$ when $v = u + w$ for a vector space where $V = U \oplus W$, what is $\text{Im}(T)$?

Say you have a vector space $V$ with an ordered basis $\mathcal{E} = [v _ 1, \ldots, v _ n]$ and another vector space $W$ with an ordered basis $\mathcal{F} = [w _ 1, \ldots, w _ m]$, and a transformation $T : V \to W$. What is the dimension of the transformation matrix $A$ that maps between them?

Say you have a vector space $V$ with an ordered basis $\mathcal{E} = [v _ 1, \ldots, v _ n]$ and another vector space $W$ with an ordered basis $\mathcal{F} = [w _ 1, \ldots, w _ m]$, and a transformation $T : V \to W$. What does the transformation matrix $A$ in $A\underline{\alpha} = \underline{\beta}$ “do” for column vectors representing $v$ and $w$ as coordinates with respect to the bases?

Translate the vector from one basis to another.

How can you still think in column vectors for any vector space?

Consider column vectors of coordinates with respect to ordered bases.

What would be the column vector $\underline{\alpha}$ representing $1 + 2x - 3x^2$ with respect to the ordered basis $[1, x, x^2]$?

Say you have a vector space $V$ with an ordered basis $\mathcal{E} = [v _ 1, \ldots, v _ n]$ and another vector space $W$ with an ordered basis $\mathcal{F} = [w _ 1, \ldots, w _ m]$, and a transformation $T : V \to W$. If $A$ is the transformation matrix, then what is the $k$-th column?

The coefficients of $T(v _ k)$ with respect to $\mathcal{F}$.