Dimension of a Vector Space

Definition

The dimension of a vector space $V$, denoted $\dim(V)$, is the number of vectors in any basis for $V$.

Key Lemma

One of the most important results in the theory of vector spaces is:

Key Lemma: Let $V$ be a vector space, and suppose that ${u_1, u_2, \ldots, u_a}$ is a set of linearly independent vectors in $V$, and that ${v_1, v_2, \ldots, v_b}$ is a spanning set for $V$. Then $a \leq b$.

Implications of the Key Lemma

1. Well-defined Dimension: All bases of a vector space have the same number of elements.
2. Extending to a Basis: Any linearly independent set can be extended to a basis.
3. Reducing to a Basis: Any spanning set can be reduced to a basis.

Examples

Dimension of $\mathbb{R}^n$

The dimension of $\mathbb{R}^n$ is $n$. The standard basis consists of the $n$ vectors:

Dimension of Polynomial Spaces

The dimension of $P_n$, the space of polynomials of degree at most $n$, is $n+1$. A basis is:

Dimension of Matrix Spaces

The dimension of $M_{m \times n}$, the space of $m \times n$ matrices, is $mn$.

Properties

1. Subspaces: If $W$ is a subspace of $V$, then $\dim(W) \leq \dim(V)$. Equality holds if and only if $W = V$.

2. Finite vs. Infinite Dimension: A vector space is said to be finite-dimensional if it has a finite basis. Otherwise, it is infinite-dimensional.

3. Sum of Subspaces: If $U$ and $W$ are subspaces of $V$, then:

4. Direct Sum: If $V = U \oplus W$ (direct sum), then:

Applications

The concept of dimension is fundamental in:

1. Solving Linear Systems: A system of $m$ linear equations in $n$ unknowns has a unique solution if and only if the dimension of the solution space is zero.

2. Linear Transformations: Understanding the dimension of the domain, range, and kernel of a linear transformation.

3. Coordinate Systems: Representing vectors in terms of basis elements.

Exercise

Prove that if ${v_1, v_2, \ldots, v_n}$ is a linearly independent set in an $n$-dimensional vector space $V$, then it is a basis for $V$.