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.

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.4: Spanning Sets And Linear Independence

Problem 74E: Let u, v, and w be any three vectors from a vector space V. Determine whether the set of vectors...

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Question

#4 #5 #6 handwritten

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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