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MATH 270 TEST 4 REVIEW
1. Let A = P DP −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 −1
1 3 −1
−1 −2 2
3. Let T : P2 → P3 be the transformation that maps a polynomial p(t) into the polynomial (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 C
2
. Find the eigenvalues and a basis for each eigenspace
in C
2
. ”
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 = P CP −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 ku + vk
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 =
−1
2
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 =
1
3
5
, u1 =
1
3
−2
, u2 =
5
1
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 R
2 have the inner product given by hx, yi = 4x1y1 + 5x2y2 3 x = (1, 1) and
y = (5, −1). Compute kxk , kyk and |hx, yi|2
.
16. Let P2 have the inner product given by evaluation at −1, 0 and 1. Compute hp, qi
where p(t) = 4 + t, q(t) = 5 − 4t
2
.
17. Based on problem 16, compute kpk and kqk.
18. For f, g ∈ C[0, 1], let hf, gi =
Z 1
0
f(x)g(x)dx. Compute h1 − 3t
2
, t − t
3
i.
19. Based on problem 18, compute kfk. (Rationalize your denominator, if necessary).
20. Find the third-order Fourier approximation to f(t) = 2π − t.
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