Lecture - Linear Algebra MT22, VIII

Flashcards

@State the dimension formula.

\[\mathrm{dim}(U + W) + \mathrm{dim}(U\cap W) = \mathrm{dim}(U) + \mathrm{dim}(W)\]

What are the three conditions for $U \oplus W = V$?

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 $V = U \oplus W$?

\[\mathrm{dim}(V) = \mathrm{dim}(U) + \mathrm{dim}(W)\]

@Define the external direct sum $U \oplus _ e V$ of $U$ and $V$.

$U \times V$ 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?