5. Basis and Dimension of Subspace

1. 키워드

Subspace(부분 공간): 선형결합에 닫혀 있는 부분 집합
Basis(기저): Fully span하며 선형 독립을 만족하는 벡터 집합
Dimension(차원): Basis의 개수
Rank(계수): Column space의 차원

2. Span and Subspace

Definition: A subspace $H$ is defined as a subset of $R^{n}$ closed under linear combination:

For any two vectors, $u_{1}$ , $u_{2} \in H$ , and any two scalars $c$ and $d$ , $c u_{1} + d u_{2} \in H$ .

$Span {v_{1}, \dots, v_{p}}$ is always a subspace, why?

$u_{1} = a_{1} v_{1} + \dots + a_{p} v_{p}, u_{2} = b_{1} v_{1} + \dots + b_{p} v_{p}$
$c u_{1} + d u_{2} = c (a_{1} v_{1} + \dots + a_{p} v_{p}) + d (b_{1} v_{1} + \dots + b_{p} v_{p}) = (c a_{1} + d b_{1}) v_{1} + \dots + (c a_{p} + d b_{p}) v_{p}$

In fact, a subspace is always represented as $Span {v_{1}, \dots, v_{p}}$ .

3. Basis of a Subspace

Definition: A basis of a subspace $H$ is a set of vectors that satisfies both of the following:

Fully spans the given subspace $H$
Linearly independent (i.e., no redundancy)

In the previous example, where $H = Span {v_{1}, v_{2}, v_{3}}$ , $Span {v_{1}, v_{2}}$ forms a plane, but $v_{3} = 2 v_{1} + 3 v_{2} \in Span {v_{1}, v_{2}}$ , ${v_{1}, v_{2}}$ is a basis of $H$ , but not ${v_{1}, v_{2}, v_{3}}$ nor ${v_{1}}$ is a basis.

4. Non-Uniqueness of Basis

Consider a subspace $H$ (green plane).

Is a basis unique?

That is, is there any other set of linearly independent vectors that span the same subspace $H$ ?

5. Dimension of Subspace

What is then unique, given a particular subspace $H$ ?

Even though different bases exist for $H$ , the number of vectors in any basis for $H$ will be unique.

We call this number as the dimension of $H$ , denoted as dim $H$ .

In the previous example, the dimension of the plane is $2$ , meaning any basis for this subspace contains exactly two vectors.

6. Column Space of Matrix

Definition: The column space of a matrix $A$ is the subspace spanned by the columns of $A$ . We call the column space of $A$ as $Col A$ .

$A = [\begin{array}{ll} 1 & 1 \\ 1 & 0 \\ 0 & 1 \end{array}]$ → $Col A = Span {[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}], [\begin{array}{l} 1 \\ 0 \\ 1 \end{array}]}$

What is dim $Col A$ ?

7. Matrix with Linearly Dependent Columns

Given $A = [\begin{array}{lll} 1 & 1 & 2 \\ 1 & 0 & 1 \\ 0 & 1 & 1 \end{array}]$ , note that $[\begin{array}{l} 2 \\ 1 \\ 1 \end{array}] = [\begin{array}{l} 1 \\ 1 \\ 0 \end{array}] + [\begin{array}{l} 1 \\ 0 \\ 1 \end{array}]$ , i.e., the third column is a linear combination of the first two.

$Col A = Span {[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}], [\begin{array}{l} 1 \\ 0 \\ 1 \end{array}], [\begin{array}{l} 2 \\ 1 \\ 1 \end{array}]}$ → $Col A = Span {[\begin{array}{l} 1 \\ 1 \\ 0 \end{array}], [\begin{array}{l} 1 \\ 0 \\ 1 \end{array}]}$