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Questions › UoH › UoH 2015 › Q8
1 mark, −0.33Mathematics › Vector spaces, rank and eigenvalues › Linear maps, rank and projectionsHardabout 2 min

UoH 2015 Question 8: Vector spaces, rank and eigenvalues

Matrix XXX of order m×nm \times nm×n is of full rank and m≠nm \neq nm=n. Let Y=XTXY = X^T XY=XTX and Z=XXTZ = X X^TZ=XXT. Let r(S)r(S)r(S) denotes rank of any matrix SSS. Then,

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Matrix X of order m × n is of full rank and m ≠ n. Let Y = Xᵀ X and Z = X Xᵀ. Let r(S) denotes rank of any matrix S. Then, (A) Y and Z are of the same rank (B) Y and Z are of the same order (C) Y and Z are inverses of each other (D) Y and Z are equal matrices

Also written as: Matrix X of order m * n is of full rank and m != n. Let Y = X^T X and Z = X X^T. Let r(S) denotes rank of any matrix S. Then,

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.

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