28.2: Four Fundamental Subspaces
- Page ID
- 69435
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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 \(m−r\)