Construct a matrix given column and null spaces

linear algebramatricessystems of equationsvector-spaces

The problem says: If possible, construct a matrix whose column space contains [1 1 0] and [0 1 1] and whose null space contains [1 0 1] and [0,0,1].

I know that the column space has to be part of the matrix, and I've managed to make two different matrices that each contain one of the null spaces with a simple system of equations, but I don't know how to go about finding both. So I'm stuck here:

\begin{bmatrix}1 & 0 & ?\\1 & 1& ?\\0&1 & ?\end{bmatrix}

Or, if its not possible, could I get an explanation?

Best Answer

As commented by Gerry Myerson, it's not possible. Suppose, towards a contradiction, that $A$ is an $m \times n$ matrix that satisfied these constraints. Then:

  • Since $[1, 1, 0] \in \textsf{Col}(A)$, we know that $m = 3$.
  • Since $[1, 1, 0]$ and $[0, 1, 1]$ are linearly independent, we know that $\textsf{dim}(\textsf{Col}(A)) \geq 2$.
  • Since $[1, 0, 1] \in \textsf{Null}(A)$, we know that $n = 3$.
  • Since $[1, 0, 1]$ and $[0, 0, 1]$ are linearly independent, we know that $\textsf{dim}(\textsf{Null}(A)) \geq 2$.

So by the rank-nullity theorem: \begin{align*} 3 &= n \\ &= \textsf{dim}(\textsf{Col}(A)) + \textsf{dim}(\textsf{Null}(A)) \\ &\geq 2 + 2 \\ &= 4 \end{align*} a contradiction.

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