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4, 2016 1. Compute these matrix multiplications: 2 1 2 1 2 1 2 2 1 2 (b) (a) 4 5 4 3 3 5 1 5 5 1 7 4 7 4 (c) 3 1 6 5 4 3 1 1 3 1 2 1 2 4 (d) 2 1 3 8...
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I have another assignment i need your help with **I don't need your assistance for #1, I've got that figured out**

Math 110 Homework Assignment 7 due date: Nov. 4, 2016 1. Compute these matrix multiplications: (a) 2 1 5 4 7 4 b 2 1 2 3 5 1 B (b) b 2 1 2 3 5 1 B  2 1 5 4 7 4 (c) b 3 1 6 5 Bb 4 3 1 1 B (d) 3 1 2 2 1 3 7 2 5 1 2 4 8 3 8 2 1 0 2. There are identities among sin and cos concerning angle addition (you may have seen them before). That is, if α and θ are two angles, there is a way to write sin( α + θ ) in terms of sin’s and cos’s of α and θ , and the same thing for cos( α + θ ). There are purely geometric proofs of these identies using triangles, but here’s a linear algebra proof: Let T α be the linear transformation from R 2 to R 2 which is rotation counterclockwise by α , and T θ the counterclockwise rotation by θ . (a) Write down the standard matrices for T α and T θ , explaining your reasoning for T α . (b) Explain what the linear transformation T α T θ does to R 2 . (c) Compute the matrix for T α T θ by multiplying the matrices for T α and T θ . (d) On the other hand, from the description in part (b), you can directly write down the matrix for T α T θ . What is that matrix? (e) Since the matrices from parts (c) and (d) are describe the same linear transforma- tion, they must be equal. What identities among sin and cos must therefore be true? (f) Using a similar idea, Fnd formulas for sin(3 θ ) and cos(3 θ ) in terms of sin( θ ) and cos( θ ). 1
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3. Suppose we have two linear transformations T 1 : R 3 −→ R 2 and T 2 : R 2 −→ R 3 given by these formulas: T 1 ( x, y, z ) = (7 x + 3 z, 2 x + y + 8 z ) and T 2 ( u, v ) = (4 u + v, 2 u + 3 v, u + 5 v ) . (a) Give the formulas for the composite function T 3 = T 2 T 1 . (b) Using these formulas, Fnd the standard matrix C for T 3 . (c) ±ind the standard matrix A for T 1 and B for T 2 . (d) Compute the matrix product BA showing the details of how you computed the entries. (You should, of course, get matrix C as an answer.) 4. Suppose that T : R n −→ R n is an invertible linear transformation (i.e., T is a bijection). This means that it’s possible to deFne the inverse function T - 1 , a function from R n to R n , by T - 1 ( Vw ) = the unique vector Vv in R n with T ( Vv ) = Vw. That certainly deFnes a function from R n to R n , but we need to check that this function is also a linear transformation. So: Prove that T - 1 is a linear transformation, using the fact that T is. Note: ±or question 4, it might help to Frst write down (to make you think about it consciously), what the ingredients are for this argument. You know (1) that T is a linear transformation, (2) that T is bijective (and hence injective and surjective), and (3) that T - 1 is the inverse function for T (and so you know the deFnition of T - 1 in terms of T ). This are the only things you know about T , and every step you make will probably involve using one of those properties. As a perhaps puzzling hint, which you’re free to ignore, if you’re ever trying to prove that two vectors Vu 1 and V u 2 are equal, it’s enough to apply T to both sides and check that T ( Vu 1 ) = T ( Vu 2 ), since T is injective. 2
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