Exercise · ZW-RANK-NULLITY-THEOREM-007 · ★★☆ Intermediate
Can there be an injective linear map \(T:\mathbb{R}^3\to\mathbb{R}^2\)?
Choose one option.
Show answer
- D. No: the image has dimension at most \(2\), so the kernel has dimension at least \(1\)
Explanation
Since \(\dim\operatorname{Im}T\le2\), the nullity \(3-\dim\operatorname{Im}T\) is at least \(1\), and a nontrivial kernel rules out injectivity. The kernel always contains zero; injective means it contains only zero.
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?
- ZW-RANK-NULLITY-THEOREM-002 A linear map T:ℝ^4→ℝ^3 is onto. What is the dimension of its kernel?
- 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?
- ZW-RANK-NULLITY-THEOREM-005 What is the dimension of the image of an injective linear map T:ℝ^3→ℝ^5?
- 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-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…
- ZW-RANK-NULLITY-THEOREM-009 If A is a 3×6 matrix, what is the smallest possible dimension of its kernel?