Exercise · ZW-RANK-NULLITY-THEOREM-010 · ★★★ Advanced
A \(5\times7\) matrix \(A\) has a kernel of dimension \(3\). What is the dimension of the space spanned by its columns?
Choose one option.
Show answer
- A. \(4\)
Explanation
The column space is the image, of dimension \(7-3=4\). It cannot be \(5\): that would require nullity \(2\).
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-007 Can there be an injective linear map T:ℝ^3→ℝ^2?
- ZW-RANK-NULLITY-THEOREM-008 The matrix C=[1 0 2 -1; 0 1 1 3; 1 1 3 2; 2 1 5 1] defines T(x)=Cx from ℝ^4 to ℝ^4. What do the dim…