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4. Let L:V-W be an isomorphism of vector space V onto vector space W. (a) Prove that L(0v) 0w where 0y and Ow are the zero vectors in V and W, respectively. (b) Show that L(v-w) L(v)-L(w). 5. Find the rank and nullity of the matrix 1 3 -2 5 4 1 4 13 5 14 2 4 3 27 -3 6 13 6. Give examples of 2 x 2 matrices such that rank (A+ B) < rank A, rank B.

4. Let L:V-W be an isomorphism of vector space V onto vector space W. (a) Prove that L(0v) 0w where 0y and Ow are the zero vectors in V and W, respectively. (b) Show that L(v-w) L(v)-L(w). 5. Find the rank and nullity of the matrix 1 3 -2 5 4 1 4 13 5 14 2 4 3 27 -3 6 13 6. Give examples of 2 x 2 matrices such that rank (A+ B) < rank A, rank B.

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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3. Let S = and T = be ordered basis for R. Let v = 5 (a) Find the coordinate vectors of v with respect to the basis S and T. (b) Find the transition matrix Ps-r from the T to the S-basis and Qr-s from the S to the T-basis. (c) Verify that Qr-s= Pr 4. Let L:V- W be an isomorphism of vector space V onto vector space W. (a) Prove that L(0v) = 0w where Oy and Oy are the zero vectors in V and W, respectively. (b) Show that L(v-w) L(v)- L(w). %3D %3D 5. Find the rank and nullity of the matrix 1 3 -2 5 4 1 4 1 3 5 14 2 4 3 27 -3 6 13 6. Give examples of 2 x 2 matrices such that rank (A+ B) < rank A, rank B.
Transcribed Image Text:3. Let S = and T = be ordered basis for R. Let v = 5 (a) Find the coordinate vectors of v with respect to the basis S and T. (b) Find the transition matrix Ps-r from the T to the S-basis and Qr-s from the S to the T-basis. (c) Verify that Qr-s= Pr 4. Let L:V- W be an isomorphism of vector space V onto vector space W. (a) Prove that L(0v) = 0w where Oy and Oy are the zero vectors in V and W, respectively. (b) Show that L(v-w) L(v)- L(w). %3D %3D 5. Find the rank and nullity of the matrix 1 3 -2 5 4 1 4 1 3 5 14 2 4 3 27 -3 6 13 6. Give examples of 2 x 2 matrices such that rank (A+ B) < rank A, rank B.
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