Math 364
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HW 12 due 5/5: 3D | 23, 24, and the following:
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A. Let \( V = P_3(\mathbb{R}) \), \( B_1 \) the standard basis, and \( B_2 = (1, x+1, (x+1)^2, (x+1)^3) \). Find the change of basis matrix from \(B_1\) to \(B_2\), and use it to write \(2x^3 - x^2 + x + 3\) in terms of \(B_2\). You may use a computer to compute a matrix inverse; just state exactly what you did.
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B. In \( \mathbb{R}^\mathbb{R} \), let \( B = (\cos^2 x, \sin^2 x, \cos x \sin x, \cos x, \sin x)\) and let \( V = Span(B) \). Let \( D: V \rightarrow V \) be differentiation. Find the matrix for \(D\) with respect to \(B\). You may assume \(B\) is a basis for \(V\).
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C. With \(B, V\) as above, let \(B' = (1, \cos x, \sin x, \cos 2x, \sin 2x)\). Find the change of basis matrix from \(B\) to \(B'\), and use it to find the matrix of \(D\) with respect to \(B'\). You may use a computer for your matrix algebra; just state exactly what you did.
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HW 11 due 4/28: 3D | 12, 18, 19, 20 (hint: restrict to \( P_m(\mathbb{R}) \rightarrow P_m(\mathbb{R})\) for the right \( m \), compute the kernel), 21, 22
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Practice problems: 3D | 2–6, 9, 11, 15
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HW 10 due 4/14: 3C | 10, 11, 12, 15, 17
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HW 9 due 4/7: 3B | 10, 13, 14, 19; 3C | 2, 4, 5
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HW 8 due 3/24: 3A | 12, 13, 15; 3B | 1, 2, 3, 6, 9
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HW 7 due 3/17: 3A | 1—4, 7, 8, 10
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Practice problems: 2C | 1, 4, 6, 8, 9, 11, 14, 18
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HW 6 due 3/3: 2B | 3—6, 8, 9
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HW 5 due 2/24: 2A | 1, 2, 5, 7, 11, 12, 13, 18
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HW 4 due 2/17: 1C | 19, 20, 21, 22, 23, 24 (Hint: Check that given any \(f\), \(\frac{1}{2}(f(x) + f(-x))\) is even.)
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HW 3 due 2/10: 1C | 10, 11, 12, 14, 15, 16, 18
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HW 2 due 2/3: 1B | 3, 8; 1C | 1, 2, 4 (for the \(b \ne 0\) case, provide a specific counterexample), 5, 6, 8
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HW 1 due 1/27: 1A | 6, 7, 12; 1B | 2, 5 (there are two directions), 6, 7
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