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MATH 51 MIDTERM 2 (MARCH 4, 2010)
Max Murphy Jonathan Campbell Jon Lee Eric Malm 11am 11am 10am 11am 1:15pm 2:15pm 1:15pm 1:15pm Xin Zhou Ken Chan (ACE) Jose Perea Frederick Fong 11am 1:15pm 11am 11am 1:15pm 1:15pm 1:15pm
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Instructions: Find your TA’s name in the table above, and circle the time that your TTh section meets. During the test, you may not use notes, books, or calculators. Read each question carefully, and show all your work. Each of the 10 problems is worth 10 points. You have 90 minutes to do all the problems.
Total
1(a). Find the inverse of the matrix A =
- Let T : R^2 → R^2 be the linear transformation defined by:
T
[
x y
]
[
x + y − 2 x + 4y
]
(a). Find the matrix A that represents the linear transformation T with respect to the standard basis S = {e 1 , e 2 }.
(b). Consider the basis B = {v 1 , v 2 } given by: v 1 =
[
]
and v 2 =
[
]
Find the change of basis matrix C for the basis B. That is, find the matrix C such that v = C[v]B for all vectors v.
(c). Find the matrix B that represents the linear transformation T with respect to the basis B.
3(a). Find all eigenvalues of the matrix A =
3(b). Consider the matrix B =
Find an eigenvector of B with eigenvalue λ = 1.
5(a). Consider a symmetric matrix B such that
B
(^) and B
Find a basis of R^3 consisting of eigenvectors of B.
5(b). If the determinant of B is 7, what are its eigenvalues? (Here B is the matrix from part (a).)
6(a).The position of a particle at time t is u(t) = (t, t^2 , t^3 ). Find the velocity of the particle at time t.
6(b). Find the acceleration of the particle at time t.
6(c). Find the speed of the particle at time t.
6(d). Find the tangent line to the path of the particle at the point (1, 1 , 1).
- Suppose F : R^3 → R^2 is defined by
F (x, y, z) =
[
xy x + 2y + sin z
]
Find the Jacobian matrix (i.e, the matrix for the total derivative) DF (1, 2 , 0).
- In part (a) and (b), find the indicated limit or else show that the limit does not exist.
9(a). lim(x,y)→(0,0)
xy x^2 + 2y^2
9(b). lim(x,y)→(0,0)
xy^2 x^2 + y^2
