Lecture - Linear Algebra MT22, IX

Flashcards

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 $T^2$?

\[T\]

@Define the vector space $\mathrm{Hom}(V, W)$.

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 $\mathrm{Ker}(T)$ of a linear map $T : V \to W$?

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

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

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

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

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

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

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

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

@State the rank-nullity theorem for a linear transformation $T : V \to W$.

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