Today we completed our introduction on quotient vector spaces and introduced a couple of important concepts:

- Linear combination & Span,
- Spanning set,
- Minimal spanning set,
- Linear independence,
- Basis & dimension.

[This is also a checklist for your learning outcomes! You have to understand all these concepts and the theory related to them.]

Below is a flowchart to help you see how/what the theory is developed.

**Remark 1.** One may try to use set operations to make subspaces. Suppose and are subspaces of a vector space. Will

(i) , (ii) , (iii)

be subspaces?

*In general* the answer is (i) NO, (ii) YES, (iii) NO.

Indeed, we have the following results:

Let and be subspaces of a vector space. Then is a subspace if and only if or .

(Proof. See Assignment 1 Q. 3.)

is a subspace.

(Proof. Exercise.)

is never a subspace.

(Proof. Exercise.)

[In view of this, you may appreciate more the definition of “Span”, which teaches you how to make subspaces.]

**Remark 2.** Recall the following two convention:

- (the zero subspace).
- Let be any subset (not necessarily finite) of a vector space. Then

**Remark 3.** We mentioned in lecture that linearly independence is a notion to characterize minimal spanning set. But how?! This is explained in the following

Claim:Let be a vector space. A linearly independent spanning set of is a minimal spanning set for .

**Proof.** Suppose is a linearly independent spanning set but not minimal.

That means we can find a *proper* subset of such that .

As , has less than elements.

i.e. The spanning set has less elements than the linearly independent set .

This contradicts to the Fundamental Thm (in p. 305 of the textbook). So is minimal.

Finally we remark the often used equivalent definitions of linearly independence/dependence:

Let be vectors in a vector space. The following are equivalent:

- are linearly independent.
- The (vector) equation
(regarding as variables) has the trivial solution only.

Below are various equivalent ways to define linearly dependent vectors.

Given vectors , then the following are equivalent:

- are linearly dependent.
- are not linearly independent.
- The equation has non-trivial solution(s).
- There exist , not all zero, such that .

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