# Lecture - Linear Algebra MT22, VI

> Source: https://ollybritton.com/notes/uni/prelims/mt22/linear-algebra/lectures/6/ · Updated: 2022-11-05 · Tags: uni, lecture

### Flashcards
What is the intuitive definition of the $\text{span}$ of a set of vectors?::
The set of all vectors reachable as linear combinations.

What’s the formal definition of $\text{span} \\{v_1, v_2, \ldots, v_k\\}$ for an underlying field $\mathbb{F}$?::
$$
\text{span} \\{v_1, v_2, \ldots, v_k\\} = \\{\sum^k_{j=1} a_jv_j, a_j\in \mathbb{F}\\}
$$

What is true (related to subspaces) about the $\text{span}$ of any set of vectors in $V$?::
The span is a vector subspace of $V$.

What do you say if $\text{span} S = V$?::
S is a “spanning set” for $V$

A set $S = \\{v_1, \ldots, v_k\\}$ is linearly independent if
$$
\sum^k_{j=1}a_j v_j = 0 \implies a_j = 0 \text{ }\forall j:1\le j\le k
$$
What’s the intuitive explanation for what this means?::
It’s impossible to express the zero vector out of non-zero scalar multiples of the vectors.

Why can’t $0 \in V$ not be a member of a set of linearly independent vectors?::
Otherwise non-zero scalar multiples could make zero.

What’s the alternative notation for $\text{span} V$?::
$$
\langle V\rangle
$$

What is true about any vector $V \in \text{span } S$ if $S$ is linearly indepedent?::
It can be written uniquely as a linear combination of the vectors in $S$.

What is a linearly indepedent spanning set for $V$ called?::
A basis.

What is true about every basis of $V$?::
It has the same number of elements.

What are the two conditions for a subspace test on $U$?::
- Show $0_V \in U$
- $\lambda u_1 + u_2 \in U$ for all $u_1, u_2, \lambda$.

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