Let V and W be the subspaces of the vector space R^4 spanned by V1= (3,-1,4,1), V2 = (5,0,5,1), V3 = (5,-5,10,3) and W1 = (9,-3,3,2), W2 = (5, -1,2,1), W3= (6,0,4,1), respectively. Find the bases and dimensions for V + W and V ∩ W, and hence prove that dim(V + W) = dim(V) + dim(W) - dim(V ∩ W).

Let V and W be the subspaces of the vector space R^4 spanned by V1= (3,-1,4,1), V2 = (5,0,5,1), V3 = (5,-5,10,3) and W1 = (9,-3,3,2), W2 = (5, -1,2,1), W3= (6,0,4,1), respectively. Find the bases and dimensions for V + W and V ∩ W, and hence prove that dim(V + W) = dim(V) + dim(W) - dim(V ∩ W).

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter4: Vector Spaces

Section4.5: Basis And Dimension

Problem 66E

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Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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