True or False. 1. If the set {v1,v2} is a basis of R^2, then the set...

True or False.

1. If the set {v1,v2} is a basis of R^2, then the set {v1,v1+v2} is also a basis of R^2.

2.If W be a vector space and V1,V2 are subspaces of W, then V1 u V2 is also a subspace of W. V1 u V2 denotes the union of V1 and V2, i.e. the set of vectors which belong to either V1 or V2 (or to both).

3.If W be a vector space and V1,V2 are subspaces of W, then V1 ^ V2 is also a subspace of W. V1 ^ V2 denotes the intersection of V1 and V2, i.e. the set of vectors which belong to both V1 and V2.

Please explain why it is true and if it is false give a counterexample.

Expert Solution

Colby Messinger answered 1 year ago

(a) If V1,V2⊂V show that (V2^⊥)⊂(V1^⊥) implies V1⊂V2 (b) If V1,V2⊂V , show that (V1+V2)^⊥=(V1^⊥)∩(V2^⊥) where...

(a) If V1,V2⊂V show that (V2^⊥)⊂(V1^⊥) implies V1⊂V2 (b) If V1,V2⊂V , show that (V1+V2)^⊥=(V1^⊥)∩(V2^⊥) where we write V1+V2 to be the subspace of V spanned by V1 and V2 .

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