Math 2250 Lab 7
Name/Unid:
Due Date: 23 October 2014
1. (a) Are the functions f1 (x) = 1, f2 (x) = cos (2x) and f3 (x) = cos2 (x) linearly independent? Why or why not?
(b) How about the functions f1 (x) = 1, f2 (x) = cos (x) and f3 (x) = cos2 (x)? Why
or why not? (Hint: Plug in 3 different values of x to get a linear system for the
coefficients in the definition of linear independence.)
2. Consider the matrix
A=
−2 11
−10 5
(a) Consider the unit vectors v1 = 15 (−3, 4)T , v2 = 51 (4, 3)T , v3 = (1, 0)T and v4 =
(0, 1)T . Compute ui = Avi for i = 1, ...4. Which ui ’s have the largest and smallest
length?
(b) Draw the vectors, vi , inside the unit circle and draw the image vectors, ui , inside
an appropriate ellipse. Label the vectors and coordinates clearly. (Hint: Recall
that an ellipse has a semi-major axis and a semi-minor axis with lengths a and
b, respectively, with a ≥ b. The distance, d, between any point on the ellipse to
the origin is such that b ≤ d ≤ a. You have been given vectors, vi , such that one
of the vectors ui lies along the semi-major axis with length a and another on the
semi-major axis with length b. Use this information to draw the ellipse.)
(c) Compute the area of the unit circle, Ac , and the area of the resulting ellipse, Ae .
Compute |det(A)| and verify that Ae = Ac |det(A)|
Page 2
3. Consider the matrix


1 0 2
A = 1 1 3
0 2 2
(a) Find a basis for the solution space of
Ax = 0
and determine the dimension of the space.
(b) Find all vectors b ∈ R3 such that
Ax = b.
has a solution. What is a basis for the space of these vectors? What is the dimension
of this space?
(c) Show that the set of basis vectors from both (a) and (b) constitute a basis for R3 .
Page 3
4. Let Mn be the vector space of all n × n matrices, with the usual operations of matrix
addition and scalar multiplication.
(a) Is the subset, W of Mn consisting of invertible matrices, A a subspace? If so, prove
it. If not, which condition(s) for a subspace does this space violate?
(b) Is the subset, W of Mn , consisting of symmetric matrices, A, with A = AT a
subspace? If so, prove it. If not, which condition(s) for a subspace does this space
violate?
(c) Find a basis for M2 . What is the dimension of this space?
(d) Find a basis for the subset of symmetric matrices in M2 . What is the dimension
of this space?
Page 4