Let U and V be subspaces of R". a) Show that UnV = {veR": 7 € U and 7 € V} is a subspace of Rn. b) Let U = null(A) and V = null(B), where A, B are matrices with n columns. Express UV as either null(C) or im(C) for some matrix C. (You may wish to write C as a block matrix.) c) Let U = null (X) where X has n columns, and V = im(Y), where Y has n rows. Show that if UnV ‡ {0} then XY is not invertible.
Let U and V be subspaces of R". a) Show that UnV = {veR": 7 € U and 7 € V} is a subspace of Rn. b) Let U = null(A) and V = null(B), where A, B are matrices with n columns. Express UV as either null(C) or im(C) for some matrix C. (You may wish to write C as a block matrix.) c) Let U = null (X) where X has n columns, and V = im(Y), where Y has n rows. Show that if UnV ‡ {0} then XY is not invertible.
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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Transcribed Image Text:Let U and V be subspaces of Rn.
a) Show that Un V
=
{√ € R¹ : √ € U and ʊ € V} is a subspace of R".
b) Let U = null(A) and V = null(B), where A, B are matrices with n columns. Express UV as either null(C)
or im(C) for some matrix C. (You may wish to write C as a block matrix.)
c) Let U = null(X) where X has n columns, and V = im(Y), where Y has n rows. Show that if UnV ‡ {0}
then XY is not invertible.
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