Exercise · ZW-RANK-NULLITY-THEOREM-008 · ★★☆ Intermediate

The matrix \(C=\begin{pmatrix}1&0&2&-1\\0&1&1&3\\1&1&3&2\\2&1&5&1\end{pmatrix}\) defines \(T(x)=Cx\) from \(\mathbb{R}^4\) to \(\mathbb{R}^4\). What do the dimensions of its kernel and image add up to?

Choose one option.

Show answer
  • B. \(4\)

Explanation

Here the rank is \(2\) and the nullity is \(2\), but the sum is always the dimension of the domain: \(4\).

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Review the concept: Rank–nullity theorem

For a linear transformation \(T\) with finite-dimensional domain \(V\), the dimension of the kernel plus that of the image equals the dimension of \(V\).

\[\dim V=\dim\ker T+\dim\operatorname{Im}T\]

More practice problems: rank–nullity theorem

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