Lecture - Linear Algebra I MT22, VI

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 $U$?