Finite-Dimensional Vector Spaces

2A. Span and Linear Independence

1) Linear Combination

Linear Combination

A linear combination of vectors $v_1,\ldots,v_m\in V$ is a vector of the form

$$a_1v_1+\cdots+a_mv_m,$$

where

$$a_1,\ldots,a_m\in\mathbb{F}.$$

Meaning: A linear combination is obtained by multiplying vectors by scalars and adding them.

2) Span

Span

The span of $v_1,\ldots,v_m$ is the set of all their linear combinations.

$$\operatorname{span}(v_1,\ldots,v_m) = \{a_1v_1+\cdots+a_mv_m \mid a_1,\ldots,a_m\in\mathbb{F}\}.$$

Meaning: The span contains every vector that can be constructed from the given vectors.

Smallest Containing Subspace

$\operatorname{span}(v_1,\ldots,v_m)$ is the smallest subspace containing

$$v_1,\ldots,v_m.$$

Meaning: Any subspace containing all the vectors must also contain their span.

Spanning List

A list $v_1,\ldots,v_m$ spans $V$ if

$$\operatorname{span}(v_1,\ldots,v_m)=V.$$

Meaning: Every vector in $V$ can be written as a linear combination of the list.

3) Finite-Dimensional Vector Space

Finite-Dimensional Vector Space

A vector space is finite-dimensional if some finite list of vectors spans the space.

Meaning: The entire vector space can be generated from finitely many vectors.

Infinite-Dimensional Vector Space

A vector space is infinite-dimensional if it is not finite-dimensional.

Meaning: No finite list of vectors can span the entire space.

4) Linear Independence

Linearly Independent

A list $v_1,\ldots,v_m$ is linearly independent if

$$a_1v_1+\cdots+a_mv_m=0$$

implies

$$a_1=\cdots=a_m=0.$$

Meaning: The zero vector can be formed only by using all-zero coefficients.

Unique Representation

If $v_1,\ldots,v_m$ is linearly independent, every vector in

$$\operatorname{span}(v_1,\ldots,v_m)$$

has a unique representation as a linear combination of these vectors.

Meaning: Linear independence prevents multiple coefficient choices from representing the same vector.

5) Linear Dependence

Linearly Dependent

A list $v_1,\ldots,v_m$ is linearly dependent if there exist scalars, not all zero, such that

$$a_1v_1+\cdots+a_mv_m=0.$$

Meaning: At least one vector contains redundant information.

Zero Vector Condition

Any list containing the zero vector is linearly dependent.

Meaning: The zero vector always creates a nontrivial linear relation.

6) Linear Dependence Lemma

Linear Dependence Lemma

If

$$v_1,\ldots,v_m$$

is linearly dependent, then for some $k$,

$$v_k \in \operatorname{span}(v_1,\ldots,v_{k-1}).$$

Removing such a $v_k$ does not change the span.

Meaning: A dependent list contains a redundant vector that can be removed.

7) Length of Independent and Spanning Lists

Length Theorem

In a finite-dimensional vector space,

$$\text{length of a linearly independent list} \le \text{length of a spanning list}.$$

Meaning: You cannot have more independent vectors than are needed to span the space.

8) Finite-Dimensional Subspaces

Finite-Dimensional Subspace

Every subspace of a finite-dimensional vector space is finite-dimensional.

Meaning: A subspace cannot require more independent directions than the whole space.

2B. Bases

1) Basis

Basis

A basis of $V$ is a list of vectors that is both

• linearly independent, and
• spanning.

Meaning: A basis contains exactly enough vectors to describe the whole space without redundancy.

2) Basis Criterion

Unique Representation Criterion

A list

$$v_1,\ldots,v_n$$

is a basis of $V$ if and only if every $v\in V$ can be written uniquely as

$$v=a_1v_1+\cdots+a_nv_n.$$

Meaning: A basis gives every vector exactly one coordinate representation.

3) Standard Basis of $\mathbb{F}^n$

Standard Basis

The standard basis of $\mathbb{F}^n$ is

$$(1,0,\ldots,0), (0,1,0,\ldots,0), \ldots, (0,\ldots,0,1).$$

Meaning: Each standard basis vector represents one coordinate direction.

4) Standard Basis of $\mathcal{P}_m(\mathbb{F})$

Polynomial Standard Basis

The standard basis of $\mathcal{P}_m(\mathbb{F})$ is

$$1,z,z^2,\ldots,z^m.$$

Meaning: Every polynomial of degree at most $m$ can be written uniquely using these powers.

5) Spanning List to Basis

Basis Reduction Theorem

Every spanning list can be reduced to a basis by removing redundant vectors.

Meaning: A spanning list may contain unnecessary vectors, which can be deleted without losing the span.

6) Existence of a Basis

Basis Existence

Every finite-dimensional vector space has a basis.

Meaning: Every finite-dimensional space has a minimal nonredundant spanning description.

7) Extending an Independent List

Basis Extension Theorem

Every linearly independent list in a finite-dimensional vector space can be extended to a basis.

Meaning: Independent vectors can always be supplemented with more vectors until they span the whole space.

8) Complementary Subspace

Complementary Subspace

If $V$ is finite-dimensional and $U$ is a subspace of $V$, then there exists a subspace $W$ such that

$$V=U\oplus W.$$

Meaning: Every subspace can be completed by another subspace to form the whole space.

2C. Dimension

1) Dimension

Dimension

The dimension of a finite-dimensional vector space $V$ is the length of any basis of $V$.

It is written as

$$\dim V.$$

Meaning: Dimension is the number of independent directions needed to describe the space.

Basis Length Theorem

Any two bases of a finite-dimensional vector space have the same length.

Meaning: Dimension does not depend on which basis is chosen.

2) Important Dimensions

Dimension of $\mathbb{F}^n$

$$\dim\mathbb{F}^n=n.$$

Meaning: $\mathbb{F}^n$ has $n$ independent coordinate directions.

Dimension of $\mathcal{P}_m(\mathbb{F})$

$$\dim\mathcal{P}_m(\mathbb{F})=m+1.$$

Meaning: A polynomial of degree at most $m$ needs $m+1$ basis polynomials.

3) Dimension of a Subspace

Subspace Dimension

If $U$ is a subspace of finite-dimensional $V$, then

$$\dim U\le\dim V.$$

Meaning: A subspace cannot have more independent directions than the whole space.

4) Independent List of the Correct Length

Independent List Basis Test

If

$$\dim V=n$$

and $v_1,\ldots,v_n$ is linearly independent, then

$$v_1,\ldots,v_n$$

is a basis of $V$.

Meaning: In an $n$-dimensional space, $n$ independent vectors automatically span the space.

5) Spanning List of the Correct Length

Spanning List Basis Test

If

$$\dim V=n$$

and $v_1,\ldots,v_n$ spans $V$, then

$$v_1,\ldots,v_n$$

is a basis of $V$.

Meaning: In an $n$-dimensional space, $n$ spanning vectors are automatically independent.

6) Full-Dimension Subspace

Full-Dimension Subspace

If $U$ is a subspace of $V$ and

$$\dim U=\dim V,$$

then

$$U=V.$$

Meaning: A proper subspace must have smaller dimension than the whole space.

7) Dimension of a Sum

Dimension Formula

For finite-dimensional subspaces $V_1$ and $V_2$,

$$\dim(V_1+V_2) = \dim V_1 + \dim V_2 - \dim(V_1\cap V_2).$$

Meaning: Shared directions are counted twice, so their dimension must be subtracted once.

8) Dimension of a Direct Sum

Direct Sum Dimension

If

$$V_1+V_2$$

is a direct sum, then

$$V_1\cap V_2=\{0\}.$$

Therefore,

$$\dim(V_1\oplus V_2) = \dim V_1+\dim V_2.$$

Meaning: Direct sums have no overlapping nonzero directions, so their dimensions simply add.