Lecture - Linear Algebra I MT22, VIII

Created: November 10, 2022

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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