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MATH 51 MIDTERM 1
October 16, 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 all solutions of the following system:
x 1 + 2 x 2 + x 3 + x 4 = 7 x 1 + 2 x 2 + 2 x 3 − x 4 = 12 2 x 1 + 4 x 2 6 x 4 = 4.
3(a) Suppose that ∆ is an equilateral triangle in R^3 and that the edges of ∆ each have length 1. Let A, B, and C be the vertices of ∆. Find
AB) · (
AC).
3(b). Suppose A =
a 1 a 2 a 3 a 4
and^ B^ =
b 1 b 2 b 3 b 4
are orthogonal vectors in^ R^4 with
a 4 > 0 and b 4 > 0.
Let a =
a 1 a 2 a 3
(^) and b =
b 1 b 2 b 3
Prove that the angle between a and b is obtuse (i.e., greater than π/2).
- Are the following three vectors in R^3 linearly independent or linearly dependent? Show your work and explain your answer.
u =
(^) v =
(^) w =
- Let
A =
6(a). What condition(s) must b satisfy to be in the column space of A?
(Your answer should be one or more equations of the form? b 1 +? b 2 +? b 3 +? b 4 = ?.)
7(a) Suppose x, y, and z are linearly independent vectors in Rn. Prove that the vectors x + y, x − y, and x + y + z are also linearly independent.
7(b) Suppose that v 1 , v 2 , v 3 , v 4 are linearly dependent vectors in Rn, and that v 1 , v 2 , v 3 are linearly independent.
Prove that v 4 is a linear combination of v 1 , v 2 , and v 3.
8(b) (4 points) Find a basis for the nullspace N (A) of A.
8(c) (3 points) Find all solutions x of
Ax =
[Hint: compare the right hand side of this equation to the columns of A.]
