# Determine whether the set of vectors are orthonormal. If the set of vectors are only orthogonal, normalize the vectors to produce an orthonormal set

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.

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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