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How do I know that the column space of the following matrix IS NOT a subspace of $R^4$? I thought that the number of columns dictated the space (which would be $R^3$)

\begin{pmatrix} 2 & -1 & 3 \\ 0 & 0 & 4 \\ 6 & -4 & 2\\ -9 & 3 & 4 \end{pmatrix}

Similarly, how do I know what the null space of the same matrix IS a subspace of $R^4$?

Please use simple language. I have trouble understanding the rules of this section of linear algebra.

Justin Keener
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1 Answers1

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Indeed column space of the given matrix is a subspace of $\mathbb R^4$ and the nullspace is a subspace of $\mathbb R^3$. More in detail since columns are linearly independent, the rank is equal to $3$, the column space has dimension $3$ and the nullspace contains only the zero vector $(0,0,0)$.

user
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  • Omg I am dumb wait I just figured it out. Column space is subspace Rm and null space is space Rn. Thank you – Justin Keener Oct 16 '20 at 18:40
  • By definition the nullspace is the space of the solutions to $Ax=0$ with $x\in\mathbb R^3$ according to the definition of matrix product. – user Oct 16 '20 at 18:42