Lecture - Linear Algebra I MT22, VIII

Created: November 10, 2022

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Flashcards

What is the dimension formula?

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$$\text{dim}(U+W)+\text{dim}(U\cap W)=\text{dim}(U)+\text{dim}(W)$$

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

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• $U,W\le V$
• $U\cap W=\{0\}$
• $U+W=V$

What is true about every $v\in V$ if $V=U\oplus W$?

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$v$ 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 $V=U\oplus W$?

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$$\text{dim}(V)=\text{dim}(U)+\text{dim}(W)$$

What’s the external direct sum of $U$ and $V$, i.e. $U{\oplus }_{e}V$?

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$U\times V$ 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?

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• $T(v_1+v_2)=T(v_1)+T(v_2)$
• $T(\alpha v_1)=\alpha T(v_1)$

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