[Math] Find the nullity and rank of a 3×5 matrix C where three columns = 0

linear algebramatricesmatrix-rank

Tricky problem. So I have a 3×5 matrix C, where {s,t,u,v,w} (I'm assuming those are its columns) is a linearly independent set of vectors in R^5, and that Cu=0, Cv=0, Cw=0.

What is the rank and nullity of C?

I'm guessing with the latter part of the question that Cu, Cv, and Cw are linearly independent in R^3, meaning the other two (Cs and Ct) are free columns, making the nullity equal to 2 and thus making the rank equal to 3. This is a wild guess, so it's probably wrong.

Best Answer

  • The elements $\{s,t,u,v,w\}$ are not the columns of $C$, they are just linearly independent vectors in $\mathbb{R}^5$.
  • The nullity of $C$ is the dimension of its nullspace, which is the subspace of $\mathbb{R}^5$ consisting of vectors $x$ satisfying $Cx=0$. You already have three linearly independent vectors in the nullspace of $C$, so the nullity is at least $3$.
  • I am not sure you can say any more. If we let your five vectors be the standard basis vectors for $\mathbb{R}^5$, then both the matrices $$C=\begin{bmatrix}0&0&0&0&0\\0&0&0&0&0\\0&0&0&0&0\end{bmatrix}$$ and $$C=\begin{bmatrix}1&0&0&0&0\\0&1&0&0&0\\0&0&0&0&0\end{bmatrix}$$ send $(0,0,1,0,0)^\top$, $(0,0,0,1,0)^\top$, and $(0,0,0,0,1)^\top$ to zero, but they have different ranks and nullities.
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