Exercise · ZW-RANK-NULLITY-THEOREM-009 · ★★★ Advanced
If \(A\) is a \(3\times6\) matrix, what is the smallest possible dimension of its kernel?
Choose one option.
Show answer
- C. \(3\)
Explanation
The rank is at most \(3\) (the number of rows), so the nullity \(6-\text{rank}\) is at least \(3\). It is attained when the three rows are independent.
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
- ZW-RANK-NULLITY-THEOREM-001 A 3×5 matrix has rank 2. What is the dimension of its kernel?
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- ZW-RANK-NULLITY-THEOREM-003 A linear map defined on a space of dimension 6 has a kernel of dimension 2. What is the dimension o…
- ZW-RANK-NULLITY-THEOREM-004 Let A=[1 2 3 4; 2 4 6 8; 1 0 1 0]. What is the dimension of its kernel?
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- ZW-RANK-NULLITY-THEOREM-006 Let T(x,y,z)=(x-y,\;y-z,\;x-z). What is the dimension of its image?
- 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…