Exercise · ZW-RANK-NULLITY-THEOREM-004 · ★☆☆ Basic
Let \(A=\begin{pmatrix}1&2&3&4\\2&4&6&8\\1&0&1&0\end{pmatrix}\). What is the dimension of its kernel?
Choose one option.
Show answer
- B. \(2\)
Explanation
The second row is twice the first and the third is independent, so the rank is \(2\). With \(4\) columns the nullity is \(4-2=2\).
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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- ZW-RANK-NULLITY-THEOREM-009 If A is a 3×6 matrix, what is the smallest possible dimension of its kernel?