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Questions › DSE › DSE 2015 › Q35
2 marks, −0.666666666666667Mathematics › Vector spaces, rank and eigenvalues › Linear maps, rank and projectionsEasyabout 2 min

DSE 2015 Question 35: Vector spaces, rank and eigenvalues

For a linear transformation P:Rn→RnP:\mathbb{R}^n\to\mathbb{R}^nP:Rn→Rn that is a projector (every xxx uniquely x=y+zx=y+zx=y+z with y∈R(P),z∈N(P)y\in\mathcal{R}(P),z\in\mathcal{N}(P)y∈R(P),z∈N(P), and P(y+z)=yP(y+z)=yP(y+z)=y), which holds?

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For a linear transformation P:ℝⁿ→ℝⁿ that is a projector (every x uniquely x=y+z with y∈R(P),z∈N(P), and P(y+z)=y), which holds? (A) P²=I (B) P=P⁻¹ (C) P²=P (D) both (a) and (b)

Also written as: For a linear transformation P:R^n->R^n that is a projector (every x uniquely x=y+z with y in R(P),z in N(P), and P(y+z)=y), which holds?

Vector spaces, rank and eigenvalues has come up 41 times in 6 exams since 2008.

What this question tests

This question is on Linear maps, rank and projections, part of Vector spaces, rank and eigenvalues in Mathematics.

Linear transformations, their range and rank, and projection operators.

TestStop has 14 past-paper questions on this subtopic.

Practise Vector spaces, rank and eigenvaluesMore on Linear maps, rank and projections

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Question from the DSE 2015 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.