28.2: Four Fundamental Subspaces

Matrix Algebra with Computational Applications · LibreTexts

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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 \(m−r\)

    This page titled 28.2: Four Fundamental Subspaces was last modified on Sat, 17 Sep 2022 15:14:37 GMT and is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Dirk Colbry via source content that was edited to the style and standards of the LibreTexts platform.