Lecture - Linear Algebra MT22, IX

If $V = U \oplus W$, and $T : V \to V$ is given by $T(v) = w$ where $v = u + w$ (uniquely), what is the name of this linear transformation?

The projection of $V$ onto $W$ along $U$.

For a linear transformation $T$ that is a projection, what is $T^2$?

\[T\]

What is the vector space $\text{Hom}(v, w)$?

The set of all linear maps from $V$ to $W$.

What is another name for an invertible linear map?

An isomorphism.

In notation, what is the kernel $\text{Ker}(T)$ of a linear map $T : V \to W$?

\[\text{Ker}(T) = \\{v \in V : T(v) = 0\\}\]

In notation, what is the image $\text{Im}(T)$ of a linear map $T : V \to W$?

\[\text{Im}(T) = \\{ w \in W : \exists v \in V \text{ s.t. } T(v) = w \\}\]

What is true about the vector spaces formed by $\text{Ker}(T)$ and $\text{Im}(T)$ for a linear transformation $T : V \to W$?

What is the nullity $n(T)$ of a linear transformation $T$?

\[\dim(\text{Ker}(T))\]

What is the rank $r(T)$ of a linear transformation $T$?

\[\dim(\text{Im(T)})\]

What is the statement of the rank-nullity theorem for a vector space $V$ and a linear transformation $T : V \to W$?

\[\dim V = n(T) + r(T)\]