1. Basis

Definition 1.1: Basis

A basis of a vector space $V$ over a field $F$ is a set $\mathcal{B} \subseteq V$ that satisfies:

Linear independence: No vector in $\mathcal{B}$ is a linear combination of the others
Spanning: $\text{span}(\mathcal{B}) = V$

Theorem 1.1: Basis Equivalences

For a finite set $\mathcal{B} = \{v_1, \ldots, v_n\}$ in a vector space $V$ , the following are equivalent:

$\mathcal{B}$ is a basis for $V$
Every $v \in V$ can be written uniquely as $v = \alpha_1 v_1 + \cdots + \alpha_n v_n$
$\mathcal{B}$ is a maximal linearly independent set
$\mathcal{B}$ is a minimal spanning set

Example 1.1: Standard Basis of ℝⁿ

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

\mathcal{E} = \{e_1, e_2, \ldots, e_n\}

where $e_i = (0, \ldots, 0, 1, 0, \ldots, 0)$ has 1 in position $i$ and 0 elsewhere.

Example 1.2: Standard Basis of Polynomials

For $P_n(F)$ = polynomials of degree ≤ $n$ , the standard basis is:

\{1, x, x^2, \ldots, x^n\}

This has $n + 1$ elements.

Theorem 1.2: Existence of Basis

Every finite-dimensional vector space has a basis. Moreover, any linearly independent set can be extended to a basis, and any spanning set can be reduced to a basis.

Proof:

Start with any spanning set (which exists since the space is finite-dimensional). If it's not independent, remove redundant vectors until it is. The resulting set is a basis.

For extension: start with the independent set, and if it doesn't span, add vectors from a spanning set using the Steinitz Exchange Lemma.

∎

Theorem 1.3: Uniqueness of Coefficients

If $\mathcal{B} = \{v_1, \ldots, v_n\}$ is a basis for $V$ , then every vector $v \in V$ has a unique representation:

v = \alpha_1 v_1 + \alpha_2 v_2 + \cdots + \alpha_n v_n

The scalars $\alpha_1, \ldots, \alpha_n$ are called the coordinates of $v$ with respect to $\mathcal{B}$ .

Proof:

Existence follows from spanning. Uniqueness: if $v = \sum \alpha_i v_i = \sum \beta_i v_i$ , then $\sum (\alpha_i - \beta_i) v_i = 0$ . By independence, $\alpha_i = \beta_i$ for all $i$ .

∎

2. Dimension

Theorem 2.1: Dimension Theorem

All bases of a finite-dimensional vector space $V$ have the same number of elements.

Proof of Theorem 2.1:

Let $\mathcal{B}_1 = \{v_1, \ldots, v_m\}$ and $\mathcal{B}_2 = \{w_1, \ldots, w_n\}$ be two bases.

Since $\mathcal{B}_1$ is independent and $\mathcal{B}_2$ spans, by the fundamental bound, $m \leq n$ .

Since $\mathcal{B}_2$ is independent and $\mathcal{B}_1$ spans, $n \leq m$ .

Therefore $m = n$ .

∎

Definition 2.1: Dimension

The dimension of a finite-dimensional vector space $V$ , denoted $\dim(V)$ , is the number of elements in any basis of $V$ .

If $V$ has no finite basis, we say $V$ is infinite-dimensional.

Example 2.1: Dimensions of Common Spaces

$\dim(\mathbb{R}^n) = n$
$\dim(P_n(F)) = n + 1$
$\dim(M_{m \times n}(F)) = mn$
$\dim(\{0\}) = 0$ (the empty set is a basis)
$\dim(F[x]) = \infty$ (infinite-dimensional)

Theorem 2.2: Dimension of Subspaces

If $W$ is a subspace of a finite-dimensional vector space $V$ , then:

$\dim(W) \leq \dim(V)$
$\dim(W) = \dim(V)$ if and only if $W = V$

Theorem 2.3: Dimension of Sum of Subspaces

If $U$ and $W$ are subspaces of a finite-dimensional vector space $V$ , then:

\dim(U + W) = \dim(U) + \dim(W) - \dim(U \cap W)

Proof:

Let $\{u_1, \ldots, u_k\}$ be a basis for $U \cap W$ . Extend it to bases $\{u_1, \ldots, u_k, v_1, \ldots, v_m\}$ for $U$ and $\{u_1, \ldots, u_k, w_1, \ldots, w_n\}$ for $W$ .

Then $\{u_1, \ldots, u_k, v_1, \ldots, v_m, w_1, \ldots, w_n\}$ spans $U + W$ and is independent (since vectors from $U$ and $W$ can only combine to zero if they're in $U \cap W$ ).

So $\dim(U + W) = k + m + n = (k + m) + (k + n) - k = \dim(U) + \dim(W) - \dim(U \cap W)$ .

∎

Corollary 2.1: Dimension of Direct Sum

If $V = U \oplus W$ , then $\dim(V) = \dim(U) + \dim(W)$ (since $U \cap W = \{0\}$ has dimension 0).

3. Direct Sums

Definition 3.1: Sum of Subspaces

Let $U, W$ be subspaces of $V$ . The sum is:

U + W = \{u + w : u \in U, w \in W\}

Definition 3.2: Internal Direct Sum

We say $V$ is the internal direct sum of subspaces $U$ and $W$ , written:

V = U \oplus W

if both conditions hold:

$V = U + W$ (sum spans $V$ )
$U \cap W = \{0\}$ (trivial intersection)

Theorem 3.1: Direct Sum Characterization

$V = U \oplus W$ if and only if every $v \in V$ can be written uniquely as:

v = u + w \quad \text{with } u \in U, w \in W

Example 3.1: Standard Decomposition of ℝ²

In $\mathbb{R}^2$ , let:

$U = \{(x, 0) : x \in \mathbb{R}\}$ (x-axis)
$W = \{(0, y) : y \in \mathbb{R}\}$ (y-axis)

Then $\mathbb{R}^2 = U \oplus W$ .

Theorem 3.2: Dimension of Direct Sum

If $V = U \oplus W$ , then:

\dim(V) = \dim(U) + \dim(W)

4. Quotient Spaces

Definition 4.1: Equivalence Relation

Define $v \sim u$ if and only if $v - u \in W$ .

This is an equivalence relation:

Reflexive: $v - v = 0 \in W$
Symmetric: $v - u \in W \implies u - v = -(v - u) \in W$
Transitive: $v - u \in W$ and $u - w \in W \implies v - w = (v - u) + (u - w) \in W$

Definition 4.2: Quotient Space

The quotient space $V/W$ is the set of all cosets:

V/W = \{v + W : v \in V\}

with vector space operations:

(v + W) + (u + W) = (v + u) + W

\alpha(v + W) = (\alpha v) + W

Theorem 4.1: Well-Definedness

The operations on $V/W$ are well-defined: they don't depend on the choice of coset representatives.

Proof of Theorem 4.1:

Suppose $v + W = v' + W$ and $u + W = u' + W$ .

Then $v - v' \in W$ and $u - u' \in W$ .

Addition: $(v + u) - (v' + u') = (v - v') + (u - u') \in W$ .

So $(v + u) + W = (v' + u') + W$ . ✓

Scalar multiplication: $\alpha v - \alpha v' = \alpha(v - v') \in W$ .

So $\alpha v + W = \alpha v' + W$ . ✓

∎

Theorem 4.2: Dimension of Quotient Space

If $W$ is a subspace of a finite-dimensional vector space $V$ , then:

\dim(V/W) = \dim(V) - \dim(W)

Proof:

Let $\{w_1, \ldots, w_k\}$ be a basis for $W$ . Extend it to a basis $\{w_1, \ldots, w_k, v_1, \ldots, v_n\}$ for $V$ .

Then $\{[v_1], \ldots, [v_n]\}$ is a basis for $V/W$ :

Spanning: Any $[v] \in V/W$ can be written $v = \sum \alpha_i w_i + \sum \beta_j v_j$ , so $[v] = \sum \beta_j [v_j]$ .
Independence: If $\sum \beta_j [v_j] = [0]$ , then $\sum \beta_j v_j \in W$ , so $\sum \beta_j v_j = \sum \alpha_i w_i$ . By independence of the full basis, all $\beta_j = 0$ .

Therefore, $\dim(V/W) = n = (k + n) - k = \dim(V) - \dim(W)$ .

∎

Example 4.1: Quotient Space Example

Let $V = \mathbb{R}^3$ and $W = \{(x, y, 0) : x, y \in \mathbb{R}\}$ (the xy-plane).

Then $V/W$ consists of all planes parallel to the xy-plane. Each such plane is a single "point" in $V/W$ .

A basis for $V/W$ is $\{[(0, 0, 1)]\}$ , so $\dim(V/W) = 1$ .

Example 4.2: Quotient by Kernel

For a linear map $T: V \to W$ , the quotient space $V/\ker(T)$ is isomorphic to $\text{im}(T)$ (this is the First Isomorphism Theorem, to be covered later).

5. Coordinate Systems and Change of Basis

Once we fix a basis, every vector has a unique coordinate representation. Changing the basis changes the coordinates, and understanding this relationship is crucial for many applications.

Definition 5.1: Coordinate Vector

Let $\mathcal{B} = \{v_1, \ldots, v_n\}$ be a basis for $V$ . For $v = \sum_{i=1}^n \alpha_i v_i$ , the coordinate vector of $v$ with respect to $\mathcal{B}$ is:

[v]_{\mathcal{B}} = \begin{pmatrix} \alpha_1 \\ \alpha_2 \\ \vdots \\ \alpha_n \end{pmatrix}

Example 5.1: Standard Coordinates

In $\mathbb{R}^3$ with standard basis $\mathcal{E} = \{e_1, e_2, e_3\}$ , the vector $(2, 3, 5)$ has coordinate vector:

[(2, 3, 5)]_{\mathcal{E}} = \begin{pmatrix} 2 \\ 3 \\ 5 \end{pmatrix}

Definition 5.2: Change of Basis Matrix

Let $\mathcal{B} = \{v_1, \ldots, v_n\}$ and $\mathcal{C} = \{w_1, \ldots, w_n\}$ be two bases for $V$ . The change of basis matrix from $\mathcal{B}$ to $\mathcal{C}$ is:

P_{\mathcal{B} \to \mathcal{C}} = \begin{pmatrix} | & | & & | \\ [v_1]_{\mathcal{C}} & [v_2]_{\mathcal{C}} & \cdots & [v_n]_{\mathcal{C}} \\ | & | & & | \end{pmatrix}

That is, the columns are the coordinate vectors of the $\mathcal{B}$ basis vectors with respect to $\mathcal{C}$ .

Theorem 5.1: Change of Coordinates Formula

For any vector $v \in V$ :

[v]_{\mathcal{C}} = P_{\mathcal{B} \to \mathcal{C}} [v]_{\mathcal{B}}

Proof:

If $v = \sum \alpha_i v_i$ , then:

[v]_{\mathcal{C}} = \sum \alpha_i [v_i]_{\mathcal{C}} = P_{\mathcal{B} \to \mathcal{C}} \begin{pmatrix} \alpha_1 \\ \vdots \\ \alpha_n \end{pmatrix} = P_{\mathcal{B} \to \mathcal{C}} [v]_{\mathcal{B}}

∎

Theorem 5.2: Properties of Change of Basis Matrix

$P_{\mathcal{B} \to \mathcal{C}}$ is invertible
$P_{\mathcal{C} \to \mathcal{B}} = (P_{\mathcal{B} \to \mathcal{C}})^{-1}$
$P_{\mathcal{B} \to \mathcal{D}} = P_{\mathcal{C} \to \mathcal{D}} P_{\mathcal{B} \to \mathcal{C}}$ (composition)

Example 5.2: Change of Basis in ℝ²

Let $\mathcal{B} = \{(1, 0), (0, 1)\}$ (standard) and $\mathcal{C} = \{(1, 1), (1, -1)\}$ .

To find $P_{\mathcal{B} \to \mathcal{C}}$ , express $(1, 0)$ and $(0, 1)$ in terms of $\mathcal{C}$ :

(1, 0) = \frac{1}{2}(1, 1) + \frac{1}{2}(1, -1), \quad (0, 1) = \frac{1}{2}(1, 1) - \frac{1}{2}(1, -1)

So:

P_{\mathcal{B} \to \mathcal{C}} = \begin{pmatrix} 1/2 & 1/2 \\ 1/2 & -1/2 \end{pmatrix}

6. Applications of Direct Sums and Quotients

Direct sums and quotient spaces are powerful tools for decomposing vector spaces and understanding their structure. They appear naturally in many areas of mathematics.

Theorem 6.1: Complementary Subspaces

For any subspace $W$ of a finite-dimensional vector space $V$ , there exists a subspace $U$ such that $V = U \oplus W$ . Such a $U$ is called a complement of $W$ .

Proof:

Let $\{w_1, \ldots, w_k\}$ be a basis for $W$ . Extend it to a basis $\{w_1, \ldots, w_k, v_1, \ldots, v_n\}$ for $V$ .

Then $U = \text{span}\{v_1, \ldots, v_n\}$ is a complement of $W$ .

∎

Remark 6.1: Non-Uniqueness of Complements

Complements are not unique. For example, in $\mathbb{R}^2$ , any line through the origin (other than the x-axis) is a complement of the x-axis.

Theorem 6.2: Isomorphism via Quotient

If $V = U \oplus W$ , then $V/W \cong U$ (isomorphic as vector spaces).

Proof:

Define $\phi: U \to V/W$ by $\phi(u) = [u]$ (the coset of $u$ ).

This is linear and bijective: every coset $[v]$ has a unique representative in $U$ (since $v = u + w$ uniquely with $u \in U, w \in W$ ).

∎

Example 6.1: Decomposing Function Spaces

Let $V = C[0, 1]$ (continuous functions) and $W = \{f : f(0) = 0\}$ .

Then $U = \{f : f \text{ is constant}\}$ is a complement of $W$ , and $V/W \cong U \cong \mathbb{R}$ .

Example 6.2: Direct Sum Decomposition of Matrices

In $M_n(F)$ , we have:

M_n(F) = \text{Sym}_n(F) \oplus \text{Skew}_n(F)

where $\text{Sym}_n(F)$ = symmetric matrices and $\text{Skew}_n(F)$ = skew-symmetric matrices.

Any matrix $A$ decomposes as:

A = \frac{A + A^T}{2} + \frac{A - A^T}{2}

Theorem 6.3: Quotient and Dimension

The quotient space $V/W$ has dimension equal to the "codimension" of $W$ in $V$ , which is $\dim(V) - \dim(W)$ . This measures how many "directions" are orthogonal (in a loose sense) to $W$ .

Example 6.3: Quotient in Linear Algebra

For solving $Ax = b$ , the solution set (if non-empty) is a coset $x_0 + \ker(A)$ in $\mathbb{R}^n/\ker(A)$ , where $x_0$ is any particular solution.

Frequently Asked Questions

Why is dimension well-defined?

The Steinitz exchange lemma shows that if you have a spanning set of size n and an independent set of size m, then m ≤ n. Applying this both ways to two bases shows they have the same size.

What's the difference between a basis and a spanning set?

A spanning set may have redundant vectors. A basis is a minimal spanning set—remove any vector and it no longer spans. Equivalently, it's a maximal independent set.

How do I find the dimension of a subspace?

Find a basis (e.g., by row reducing the matrix whose rows generate the subspace) and count the basis vectors. Or use the rank of the associated matrix.

What's the intuition for quotient spaces?

V/W 'collapses' W to a point. Elements of V/W are parallel copies of W. If W is a line through origin, V/W contains all lines parallel to W, each as a single 'point' in the quotient.

How do I verify that V = U ⊕ W?

Check two things: (1) U + W = V (every vector is a sum), and (2) U ∩ W = {0} (only overlap is zero). Equivalently, show every v has a UNIQUE decomposition as u + w.

Basis & Dimension Practice

10

Questions

0

Correct

0%

Accuracy

1

What is

\dim(\mathbb{R}^5)

?

Easy

Not attempted

2

What is

\dim(M_{2 \times 3}(\mathbb{R}))

?

Easy

Not attempted

3

What is

\dim(\mathbb{R}_3[x])

(polynomials of degree ≤ 3)?

Easy

Not attempted

4

If

\dim(V) = n

, how many vectors are in any basis of

V

?

Medium

Not attempted

5

If

V = U \oplus W

, what is

U \cap W

?

Easy

Not attempted

6

If

\dim(V) = 5

and

\dim(W) = 2

where

W \subseteq V

, what is

\dim(V/W)

?

Easy

Not attempted

7

Can 3 vectors form a basis for

\mathbb{R}^4

?

Medium

Not attempted

8

If

U

and

W

are subspaces with

V = U + W

and

\dim(U \cap W) = 0

, then:

Medium

Not attempted

9

What is

[v] + [u]

in

V/W

?

Easy

Not attempted

10

If

V = U \oplus W

and

\dim U = 3

,

\dim W = 4

, what is

\dim V

?

Easy

Not attempted