Explanation Given: Two subspaces V and W are complements Formula Used: Two subspaces V and W of R n are complements if any vector x → in R n can be expressed uniquely as x → = v → + w → , where v → is in V and w → is in W Calculation: Let us assume that y → ∈ V ∩ W Given that V and W are complements Hence, any vector y → in R n can be expressed uniquely as x → = v → + w → , where v → is in V and w → is in W Now, y → can be expressed as follows: y → = y → + 0 , where y → = 1 4 y → + 3 4 y → , such that y → , 0 , 1 4 y → + 3 4 y → ∈ V , W Thus, vector y → can never be expressed uniquely as the sum of v → and w → Hence, V ∩ W = [ 0 → ] Also, let us assume that { v → 1 , v → 2 , ... v → k } and { w → 1 , w → 1 , ... w → p } are the basis of V and W respectively Let B = { v → 1 , v → 2 , ... v → k , w → 1 , w → 1 , ... w → p } Now subspaces V and W of R n are complements Thus, vector x → in R n can be expressed uniquely as x → = v → + w → , where v → is in V and w → is in W And, here x → lies in s p a n B Therefore, s p a n B = R n Now, check that B is independent Assume that a linear combination of vectors v → 1 , v → 2 ,

Linear Algebra with Applications (2-Download)

5th Edition

ISBN: 9780321796974

Author: Otto Bretscher

Publisher: PEARSON

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Chapter 3.3, Problem 60E

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To show that two subspaces V and W are complements if and only if V∩W=[0→] and dim(V)+dim(W)=n

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Chapter 3.3, Problem 59E

Chapter 3.3, Problem 61E

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Chapter 3 Solutions

Linear Algebra with Applications (2-Download)

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To show that two subspaces V and W are complements if and only if V ∩ W = [ 0 → ] and dim ( V ) + dim ( W ) = n

Solution Summary: The author explains that two subspaces V and W are complements if and only when they are expressed as stackrel

To show that two subspaces V and W are complements if and only if V∩W=[0→] and dim(V)+dim(W)=n

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Chapter 3 Solutions