Four Fundamental Subspaces

#Linear Algebra

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1. Column space of \(A\)
2. Row space of \(A\) (column space of \(A^T\))
3. Null space of \(A\)
   • The null space of \(A\) consists of all solutions of \(A\boldsymbol{x} = \boldsymbol{0}\), denoted by \(N(A)\)
4. Left null space of \(A\) (null space of \(A^T\))

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Column Space

Define:

All linear combinations of the columns of \(A\).
All solution(s) to \(A\boldsymbol{x} = \boldsymbol{b}\).

• Denoted by \(C(A)\)

Null Space

Define:

All solution(s) to \(A\boldsymbol{x} = \boldsymbol{0}\).

• Denoted by \(N(A)\)

Theorem

Suppose \(A \in \mathbb{R}^{m\times n}\)

• \(C(A)\) is a subspace of \(\mathbb{R}^m\)
• \(N(A)\) is a subspace of \(\mathbb{R}^n\)

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Calulation

Finding Column Space

Finding Null space

Define:

variable corresponding to pivot column: pivot variable

variable corresponding to free column (non-pivot column): free variable

1. Use Gaussian-Jordan elimination: \(A \rightarrow R\) (RREF)
2. Identify pivot variables and free variables.
3. Identify special solutions.
4. \(N(A)\) = \(span(\{special\ solutions\})\)

If there are no free variables in step 2, \(N(A)=\{\boldsymbol{0}\}\). In this case, \(dim(N(A))=0\). (因為 basis 是 \(\{\emptyset\}\))

The dimension of null space

Define:

The dimension of a null space is the number of free variables, i.e.

\(dim(N(A))\) = number of special solutions