# hw4 - MATH 223 HOMEWORK#4 Due Friday Oct 12 Problem 1 2 in...

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MATH 223 - HOMEWORK #4 Due Friday, Oct 12 Problem 1. 2 in Section 2.1. Problem 2. 5 in Section 2.1. Problem 3. 10 in Section 2.1. (Note that if we know T ( ~v ) and T ( ~w ), then we also know T ( α~v + β ~w ) for any α, β R .) Problem 4. 17 in Section 2.1. Problem 5. 18 in Section 2.1. (This is a bit confusing because usually null-spaces and ranges lie in diﬀerent spaces, in V and W , respectively. In this case V = W , so they lie in the same space. Note that T is determined once we know what T ( ~e 1 ) and T ( ~e 2 ) are. Choose these two vectors so that the trasnformation is as required.) Problem 6. Recall the deﬁnition of a vector space V in Homework #1. V = { ~a | a R , a > 0 } , ~a + ~ b = --→ a · b, c · ~a = -→ a c . Consider the function E : R 1 V deﬁned by E ( x ) = -→ e x . (1)Show that E is a linear transformation. (2)Note that E is both one-to-one and onto. Describe the inverse E - 1 : V R 1 . Problem

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hw4 - MATH 223 HOMEWORK#4 Due Friday Oct 12 Problem 1 2 in...

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