Linear Algebra and Its Applications · Problem 2.4.21

Vector Spaces — Gilbert Strang

Mathematics · four-fundamental-subspaces

Problem

Construct a matrix with the required property, or explain why you can't. (a) Column space contains [110]\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, [001]\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix}, row space contains [12]\begin{bmatrix} 1 \\ 2 \end{bmatrix}, [25]\begin{bmatrix} 2 \\ 5 \end{bmatrix}. (b) Column space has basis [123]\begin{bmatrix} 1 \\ 2 \\ 3 \end{bmatrix}, nullspace has basis [321]\begin{bmatrix} 3 \\ 2 \\ 1 \end{bmatrix}. (c) Dimension of nullspace =1+= 1 + dimension of left nullspace. (d) Left nullspace contains [13]\begin{bmatrix} 1 \\ 3 \end{bmatrix}, row space contains [31]\begin{bmatrix} 3 \\ 1 \end{bmatrix}. (e) Row space == column space, nullspace ≠\neq left nullspace.

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