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Math270Test4ReviewKeySpring2020.pdf
MATH 270 TEST 4 REVIEW KEY
1. Let A = PDP−1 and compute A4 where P =
[ 5 7
2 3
] and D =
[ 2 0
0 1
] .
Answer:
[ 226 −525 90 −209
]
2. Diagonalize the following matrix where the eigenvalues are λ = 5,1. 2 2 −11 3 −1 −1 −2 2
Answer:
5 0 00 1 0
0 0 1
3. Let T : P2 → P3 be the transformation that maps a polynomial p(t) into the polyno- mial (t + 5)p(t). Find the matrix for T relative to the bases {1, t, t2} and {1, t, t2, t3}.
Answer :
5 0 0
1 5 0
0 1 5
0 0 1
4. Let the following matrix act on C2. Find the eigenvalues and a basis for each eigenspace in C2. [
1 5
−2 3
]
Answer : λ = 2±3i ; v1 =
[ 1−3i
2
] , v̄2 =
[ 1 + 3i
2
]
5. Find an invertible matrix P and a matrix C of the form
[ a −b b a
] 3
the given matrix has the form A = PCP−1. Use the information from problem 4.[ 1 5
−2 3
]
Answer : P =
[ 1 3
2 0
] ,C =
[ 2 −3 3 2
]
6. Find the distance between x =
[ 10
−3
] and y =
[ −1 −5
] .
Answer : 5 √ 5
7. Let u =
2−5 −1
and v =
−7−4
6
. Compute‖u + v‖2 .
Answer : 131
8. Compute the orthogonal projection of
[ 1
7
] onto the line through
[ −4 2
] and the origin.
Answer :
[ −2 1
]
9. Let y =
[ 2
3
] and u =
[ 4
−7
] . Write y as the sum of two orthogonal vectors, one in Span{u}
and one orthogonal to u.
Answer : y =
−
4 5
7 5
+
14 5
8 5
10. Let y =
[ 3
1
] and u =
[ 8
6
] . Compute the distance from y to the line through u and the
origin.
Answer : 1
11. Determine whether the set of vectors are orthonormal. If the set of vectors are only
orthogonal, normalize the vectors to produce an orthonormal set.
(Rationalize your denominator, if necessary).
1 3
1 3
1 3
,
−1
2
0
1 2
Answer :
√ 3 3
√ 3 3
√ 3 3
,
− √ 2 2
0
√ 2 2
12. Find the orthogonal projection of y onto the Span{u1, u2}.
y =
−12
6
, u1 =
3−1
2
, u2 =
1−1 −2
Answer : y =
−12
6
13. Let W be the subspace spanned by the u’s and write y as the sum of a vector in W
and a vector orthogonal to W.
y =
13
5
, u1 =
13 −2
, u2 =
51
4
Answer : y =
10 3
2 3
8 3
+
−7
3
7 3
7 3
14. Find an orthogonal basis for the column space of the following matrix.
3 −5 1 1 1 1
−1 5 −2 3 −7 8
Answer :
3
1
−1 3
,
1
3
3
−1
,
−3 1
1
3
15. Let R2 have the inner product given by 〈x, y〉 = 4x1y1 + 5x2y2 3 x = (1,1) and y = (5,−1). Compute ‖x‖ ,‖y‖ and |〈x, y〉|2.
Answer : ‖x‖ = 3, ‖y‖ = √ 105, |〈x, y〉|2 = 225
16. Let P2 have the inner product given by evaluation at −1,0 and 1. Compute 〈p,q〉 where p(t) = 4 + t, q(t) = 5−4t2.
Answer : 28
17. Based on problem 16, compute ‖p‖ and ‖q‖.
Answer : ‖p‖ = 5 √ 2 ; ‖q‖ = 3
√ 3
18. For f, g ∈ C[0,1], let 〈f,g〉 = ∫ 1 0
f(x)g(x)dx. Compute 〈1−3t2, t− t3〉.
Answer : 0
19. Based on problem 18, compute‖f‖. (Rationalize your denominator, if necessary).
Answer : 2 √ 5
5
20. Find the third-order Fourier approximation to f(t) = 2π − t.
Answer : π + 2 sin(t) + sin(2t) + 2
3 sin(3t)
