MAT247H1 Lecture Notes - Lecture 1: Kingdom Of Sine, Linear Map, Joule
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Problem 1: it is straightforward to verify this is an inner product, again, easily veri ed using matrix multiplication. 1 4x2: not an inner product. 2, take x1 = 5 and x2 = 1, we see that < , > is not positive de nite. Hence, not an inner product: not an inner product. Take f (x) = x(x i) then < f, f >= 0, but f (cid:54)= 0, hence not an inner product: easy to verify it is an inner product. For condition 4, we use the property that < , >(cid:48) is an inner product, hence we get positive de niteness. (cid:80)n. In this question, to be precise, we need to make sure the de nition is well- de ned. , vn}, with vi"s in (i. e x = i=1 aivi), then for any other basis vector w , the only way to represent x as an element of span{v1, .
