Lecture - Linear Algebra I MT22, X

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?

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

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?