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Questions › UoH › UoH 2021 › Q34
1 mark, −0.33Mathematics › Vector spaces, rank and eigenvalues › Linear maps, rank and projectionsHardabout 2 min

UoH 2021 Question 34: Vector spaces, rank and eigenvalues

For an mmm by nnn matrix S having full column rank, m>nm > nm>n, let X=STSX = S^T SX=STS and Y=SSTY = S S^TY=SST. Then, which of the following are true?

I. X and Y are both of the same order. II. X and Y are both of the same rank. III. X and Y are both symmetric matrices. IV. X and Y are both non-singular.

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For an m by n matrix S having full column rank, m > n, let X = Sᵀ S and Y = S Sᵀ. Then, which of the following are true? I. X and Y are both of the same order. II. X and Y are both of the same rank. III. X and Y are both symmetric matrices. IV. X and Y are both non-singular. (A) I and II are true (B) II and III are true (C) III and IV are true (D) I and IV are true

Also written as: For an m by n matrix S having full column rank, m > n, let X = S^T S and Y = S S^T. Then, which of the following are true? I. X and Y are both of the same order. II. X and Y are both of the same rank. III. X and Y are both symmetric matrices. IV. X and Y are both non-singular.

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 UoH 2021 paper. Solutions are TestStop's own. Not affiliated with or endorsed by any examination authority.