hw3 - HOMEWORK 3 1 Let U = x y x,y ∈ R E R 3 Suppose T U...

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Unformatted text preview: HOMEWORK 3 1. Let U = x y : x,y ∈ R E R 3 . Suppose T : U → R is linear. (a) Show that if T is not the zero transformation, then the map T : R 3 → R defined by: T ( x ) = T ( x ) x ∈ U x ∈ R 3 \ U is not a linear transformation. (b) Find a map T : R 3 → R such that T U = T which is linear. [Hint: Find a way to apply the Theorem on Extending by Linearity.] (c) Define Ψ : L ( R 3 , R ) → L ( U, R ) by Ψ( T ) = T U . Show that this is a surjective linear transformation. 2. Can you think of a function t : R → R such that ∀ x,y ∈ R , t ( x + y ) = t ( x ) + t ( y ), but it’s not the case that ∀ a ∈ R ∀ x ∈ R , t ( ax ) = at ( x )? That is, a function on the reals which is additive but which isn’t a linear tranformation when R is regarded as a vector space over itself? We’ll find such a function in this exercise. • We know R is a vector space over the field R , but convince yourself that R is also a vector space over the field...
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hw3 - HOMEWORK 3 1 Let U = x y x,y ∈ R E R 3 Suppose T U...

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