math test
MATH 270 TEST 4 REVIEW
1. Let A = PDP−1 and compute A4 where P =
[ 5 7
2 3
] and D =
[ 2 0
0 1
] .
2. Diagonalize the following matrix where the eigenvalues are λ = 5,1. 2 2 −11 3 −1 −1 −2 2
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}.
4. Let the following matrix act on C2. Find the eigenvalues and a basis for each eigenspace in C2. [
1 5
−2 3
]
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
]
6. Find the distance between x =
[ 10
−3
] and y =
[ −1 −5
] .
7. Let u =
2−5 −1
and v =
−7−4
6
. Compute‖u + v‖2 .
8. Compute the orthogonal projection of
[ 1
7
] onto the line through
[ −4 2
] and the origin.
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.
10. Let y =
[ 3
1
] and u =
[ 8
6
] . Compute the distance from y to the line through u and the
origin.
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
12. Find the orthogonal projection of y onto the Span{u1,u2}.
y =
−12
6
, u1 =
3−1
2
, u2 =
1−1 −2
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
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
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.
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.
17. Based on problem 16, compute ‖p‖ and ‖q‖.
18. For f, g ∈ C[0,1], let 〈f,g〉 = ∫ 1 0
f(x)g(x)dx. Compute 〈1−3t2, t− t3〉.
19. Based on problem 18, compute‖f‖. (Rationalize your denominator, if necessary).
20. Find the third-order Fourier approximation to f(t) = 2π − t.