# Suppose that 51.0 mL of 0.11 M AgNO3 is added to 50.0 mL of 0.048 M…

1. Let V be a 2-dimensional vector space with basis X = {v1, v2}, write down the matrices [0]xx and [id]xx. 2. Let U, V, W be vector spaces and S:U +V, T:V + W be linear transforma- tions. Define the composition TOS:U + W by To S(u) = T(S(u)) for all u in U. a. Show that ToS is a linear transformation. b. Now suppose U is 1-dimensional with basis X {41}, V is 2-dimensional with basis Y = {V1, V2}, W is 3-dimensional with basis 3 = {W1, W2, W3}. Show (To S]3x = [T]39[S]yx. This equality holds for compositions of linear transformations in general, and this is why we defined matrix multiplication the way we did. 3. Let T : P2 _ Pi be the linear transformation given by differentiation and S:P1 _ P2 be the linear transformation given by S(p(x)) = S5p(t)dt for all p(x) in P1. Let X = {1,x} and Y = {1, x, x2}. a. Write down the matrices (T]xy and [S]yx. b. Compute (T]xy [S]yx and [S]yx[T]xy. Conclude that Tos = idp,. Note how this is consistent with the fact that differentiation after integration gives back the original function. Why is SoT not idp,?

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