28.2: Four Fundamental Subspaces
- Page ID
- 69435
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The four fundamental subspaces
- Columnspace, \(\mathcal{C}(A)\)
- Nullspace, \(\mathcal{N}(A)\)
- Rowspaces, \(R(A)\)
- All linear combinations of rows
- All the linear combinations of the colums of \(A^\top\), \(\mathcal{C}(A^\top)\)
- Nullspace of \(A^\top\), \(\mathcal{N}(A^\top)\) (the left nullspace of \(A\))
Where are these spaces for a \(m \times n\) matrix \(A\)?
- \(\mathcal{R}(A)\) is in \(R^n\)
- \(\mathcal{N}(A)\) is in \(R^n\)
- \(\mathcal{C}(A)\) is in \(R^m\)
- \(\mathcal{N}(A^\top)\) is in \(R^m\)
Calculating basis and dimension
For \(\mathcal{R}(A)\)
- If \(A\) undergoes row reduction to row echelon form \(B\), then \(\mathcal{C}(B)\neq \mathcal{C}(A)\), but \(\mathcal{R}(B) = \mathcal{R}(A)\) (or \(\mathcal{C}(B^\top) = \mathcal{C}(A^\top)\))
- A basis for the rowspace of \(A\) (or \(B\)) is the first \(r\) rows of \(B\)
- So we row reduce \(A\) and take the pivot rows and transpose them
- The dimension is also equal to the rank \(r\)
For \(\mathcal{N}(A)\)
- The bases are the special solutions (one for every free variable, \(n−r\))
- The dimension is \(n−r\)
For \(\mathcal{C}(A) = \mathcal{R}(A^\top)\)
- Apply the row reduction on the transpose \(A^{\top}\).
- The dimension is the rank \(r\)
For \(\mathcal{N}(A^\top)\)
- It is also called the left nullspace, because it ends up on the left (as seen below)
- Here we have \(A^\top y = 0\)
- \(y^\top(A^\top)^\top = 0^\top\)
- \(y^\top A = 0^\top\)
- This is (again) the special solutions for \(A^{\top}\) (after row reduction)
- The dimension is \(m−r\)