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Note: N(A) denotes the null space of the matrix A, while R(A) stands for the range of A, viz., the column space of A. = 1. (a) Let matrices A, B, C be such that A R(B), whereas N(C) ≤ N(A). BC. Show that R(A) C (b) Show that N(ATA) N(A) for any matrix A, even rectangular. (Hint: Ax = 0 iff || Ax ||= 0.) (c) Using (b), show that R(ATA) = R(AT). =

Note: N(A) denotes the null space of the matrix A, while R(A) stands for the range of A, viz., the column space of A. = 1. (a) Let matrices A, B, C be such that A R(B), whereas N(C) ≤ N(A). BC. Show that R(A) C (b) Show that N(ATA) N(A) for any matrix A, even rectangular. (Hint: Ax = 0 iff || Ax ||= 0.) (c) Using (b), show that R(ATA) = R(AT). =

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Matrix Analysis parctice question, please show clear, thanks

Note: N(A) denotes the null space of the matrix A, while R(A) stands for the range of A, viz., the column space of A. 1. (a) Let matrices A, B, C be such that A R(B), whereas N(C) ≤ N(A). (b) Show that N(ATA) (Hint: Ax = = 0 iff || Ax ||= 0.) = N(A) for any matrix A, even rectangular. BC. Show that R(A) ≤ = (c) Using (b), show that R(ATA) = R(AT). = (Hint 1. Let A, B be matrices such that R(A) ≤ R(B). If rk(A) = rk(B), then show that the subspaces are equal. Hint 2. rank + nullity dimension of codomain).
Transcribed Image Text:Note: N(A) denotes the null space of the matrix A, while R(A) stands for the range of A, viz., the column space of A. 1. (a) Let matrices A, B, C be such that A R(B), whereas N(C) ≤ N(A). (b) Show that N(ATA) (Hint: Ax = = 0 iff || Ax ||= 0.) = N(A) for any matrix A, even rectangular. BC. Show that R(A) ≤ = (c) Using (b), show that R(ATA) = R(AT). = (Hint 1. Let A, B be matrices such that R(A) ≤ R(B). If rk(A) = rk(B), then show that the subspaces are equal. Hint 2. rank + nullity dimension of codomain).
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