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hw06 - RS Hint such a vector v satisfies the equation RS v...

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Math 110—Linear Algebra Fall 2009, Haiman Problem Set 6 Due Monday, Oct. 12 at the beginning of lecture. 1. Section 2.4, Exercise 9. 2. Section 2.4, Exercise 16. 3. Prove or disprove the following statement: the set of invertible linear transformations from V to W is a subspace of L ( V, W ). 4. Let R be the rotation in R 3 about the x -axis, by π/ 4 in the direction from the y -axis towards the z -axis. Let S be the rotation in R 3 about the z -axis, by π/ 4 in the direction from the x -axis toward the y -axis. (a) Find the matrices with respect to the standard basis in R 3 of R , S and RS . (b) Assuming that RS is also a rotation (in fact, it is true that the composite of any two
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Unformatted text preview: RS . Hint: such a vector v satisfies the equation RS ( v ) = v . 5. Let A and B be the matrices of the rotations R and S in Problem 2. Find a change of coordinate matrix Q such that B = Q-1 AQ . 6. Let V be a finite dimensional vector space. Let α , β , γ and δ be ordered bases of V . (a) If the change of coordinate matrices [ I ] β α and [ I ] δ γ are equal, does it follow that α = γ and β = δ ? (b) If [ I ] β α = [ I ] γ α , does it follow that β = γ ? Justify your answers....
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