Download Linear Map - Linear Algebra and Multivariable Calculus - Second Midterm Exam and more Exams Calculus in PDF only on Docsity!
MATH 51 MIDTERM 2
November 13, 2008
Professor: Han Kargin White Wise TTh Section Number:
TA:
Olena Bormashenko Luis Diogo Kaveh Fouladgar Frederick Tsz Ho Fong Robin Koytcheff Jason Lo Jonathan Lee Jose Perea Josh Genauer
Time your TTh section meets: morning afternoon
Your name (print): Student ID:
Sign to indicate that you accept the honor code:
Instructions: Circle your professor’s name, your TA’s name, and the time that you attend the TTh section. During the test, you may not use notes, books, or calculators. Read each question carefully, and show all your work. Each of the nine problems is worth 10 points. You have 90 minutes to do all the problems.
Total
- Find the following.
1(a). The matrix for for the linear map T given by T
([
x y
])
7 x − 2 y x + 3y 5 y
1(b). The matrix for reflection in R^2 about the line y = −x.
1(c). The matrix for T : R^3 → R^3 , where T is rotation by 180◦^ about the y-axis, followed by rotation by 180◦^ about the z-axis.
- Find the inverse of the matrix A =
4(a). Find all eigenvalues of the matrix A =
4(c). Suppose that C is a symmetric 2x2 matrix with determinant 7. Suppose that the vector
v =
[
]
is an eigenvector of C with eigenvalue 5. Find an eigenvector w that is not a scalar
multiple of v, and find its eigenvalue. Explain.
- Consider the basis B of R^2 consisting of v 1 =
[
]
and v 2 =
[
]
5(a). Find the matrix C such that w = C[w]B for all vectors w ∈ R^2.
5(b). Find a matrix M so that [w]B = M w for all w ∈ R^2.
- Let V be the subspace of R^3 spanned by v 1 =
(^) and v 2 =
Consider the coordinate system for V determined by the basis B = {v 1 , v 2 }.
6(a). Find the vector w ∈ R^3 whose expression in the B-coordinate system is
[w]B =
[
]
6(b). Find [v]B (the expression for v in the B-coordinate system) for the vector
v =
- Suppose that F : R^2 → R^2 is a map such that F (0, 0) =
[
]
and such that DF (0, 0) =
[
]
8(a). Estimate F (. 002 , .003).
8(b). Find a point (x, y) near (0, 0) so that F (x, y) '
[
]
9(a). Suppose v is an eigenvector of A with eigenvalue λ. Prove that v is also an eigenvector of I + A^2 , and find its eigenvalue.
9(b). Suppose that A and B are similar matrices, i.e., that B = C−^1 AC for some invertible matrix C. Suppose that λ is a real number. Prove that λI − A and λI − B are also similar.
