# Lecture - Linear Algebra MT22, IX

> Source: https://ollybritton.com/notes/uni/prelims/mt22/linear-algebra/lectures/9/ · Updated: 2022-11-19 · Tags: uni, lecture

### 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 $T^2$?::
$$
T
$$

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 $T : V \to W$?::
- $\text{Ker}(T) \le V$
- $\text{Im}(T) \le W$

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 $T : V \to W$?::
$$
\dim V = n(T) + r(T)
$$

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