What is the dimension of the image of an injective linear map \(T:\mathbb{R}^3\to\mathbb{R}^5\)?

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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 \(V\).

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-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…
• ZW-RANK-NULLITY-THEOREM-009 If A is a 3×6 matrix, what is the smallest possible dimension of its kernel?

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