## Transcribed Text

1. The augmented matrix of a linear system has the form
1 2 1
a
-1 3 1 1
b
3 -5 5 1
C
2 -2 4 2
d
(a) Can you determine by inspection whether the determinant of the coefficient
matrix is 0? Explain.
(b) Can you decide by inspection whether the linear system has a unique solu-
tion for every choice of a, b, C. d? Explain.
(c) Determine the values of a, b, c. d for which the linear system is consistent
and inconsistent.
(d) If a - 2, b = 1. C - -1, and d - 4, find the solution set for the linear
system.
2. Let
1
1
M1 =
1
M2 =
2
1
M3 = 1 :
(a) Show that the set {M1. M2, Ms} is linearly independent.
(b) Describe all matrices that can be written as a linear combination of M1, M2,
and M3.
(c) Find a basis for the space of matrices found in part (b) and give the space's
dimension.
(d) Give an example of a space that is isomorphic to the space found in (b).
Specify the isomorphism between the two spaces.
3. Let c be a fixed scalar and let
m(x) 1
p((c)==to
(3(1)=(I+C)? =
(a) Show that B - {p1(x) p2(x), P3(x)} is a basis for P2.
(b) Find the coordinates of f(x) = ao + ax + 02x² relative to B.
4. Let W C R3 (i.e. W is a subset of R³) be the plane with equation I- =
21
i.e. W = {F € R3 on - T2 - T3 - 0} where Fil =
T2
T3
(a) Show W is a subspace of the vector space R³ with standard addition and
scalar multiplication
(b) Find a basis for W and then use the Gram-Schmidt procedure to obtain an
orthonormal basis for W.
5. Define a transformation T P2 R by
T(p(x)) = fo p(c)de
(a) Show T is a linear transformation
(b) Compute N(T). Is T one-to-one?
(c) Show that T is onto.
(d) Let B be the standard basis for P2 and let B' = {1} be a basis for R. Find
[T),
(e) Use the matrix found in part (d) to compute T1-x2 - - 3x+2).
a
6. Let A =
for some a. 6 € R.
b
a
(a) Find the eigenvalues and eigenvectors of A.
(b) Diagonalize A. That is, specify P and D such that D = P-1AP.
7. Let A be an n x n matrix. Show that det(A7) = det(A) using a proof by
induction. (Hint: the base case is not n=1.)
8. Suppose that S = {01,02,23} is linearly independent and
W1 = is + in + is
12-
w3 U3.
Show that T - {ur W2 w3} is linearly independent.

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