# Lecture - Linear Algebra MT22, VIII

> Source: https://ollybritton.com/notes/uni/prelims/mt22/linear-algebra/lectures/8/ · Updated: 2022-11-10 · Tags: uni, lecture

### Flashcards
What is the dimension formula?::
$$
\text{dim}(U + W) + \text{dim}(U\cap W) = \text{dim}(U) + \text{dim}(W)
$$

What are the three conditions for $U \oplus W = V$?::
- $U, W \le V$
- $U \cap W = \\{0\\}$
- $U + W = V$

What is true about every $v \in V$ if $V = U \oplus W$?::
$v$ can be uniquely written as $v = u + w$ with $u \in U$ and $w \in W$.

What is a consequence of the dimension formula if $V = U \oplus W$?::
$$
\text{dim}(V) = \text{dim}(U) + \text{dim}(W)
$$

What’s the *external* direct sum of $U$ and $V$, i.e. $U \oplus_e V$?::
$U \times V$ equipped with componentwise addition and scalar multiplication.

If $V, W$ are vector spaces over $\mathbb{F}$ what are the two conditions for a function $T: V \to W$ to be a linear transformation/map?::
- $T(v_1 + v_2) = T(v_1) + T(v_2)$ 
- $T(\alpha v_1) = \alpha T(v_1)$

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