Math 212 Final Exam Solutions
(1) (a) Let B denote the standard basis, and B
20
2 1
[T ]B =
1 2
= cfw_v1 , v2 , v3 . Then by denition:
1
0
1
(b) By matrix multiplication:
3
2
[T v1 ]B = 2 [T v2 ]B = 1
0
1
1
[T v3 ]B = 1
3
Using the fact that:
a+b
[av1 + b
Math 212 Quiz # 5 April 22, 2005
(1) Write down both the cofactor and permutation expansions of the determinant of an
n n matrix. Explain your notation. Solution.
n
(1)i+j Aij det Mij
det A =
j =1
sgn( )A1(1) An(n)
det A
Sn
where A = (Aij ) is the matrix
Math 212 Quiz # 6 May 4, 2005
(1) State the Cayley-Hamilton Theorem. Solution. If T : V V is a linear transformation of a nite dimensional vector, and p(x) is the characteristic polynomial of T ,
then p(T ) = 0.
(2) Answer the following True or False. Be
110.212 Quiz V Solution
April 21, 2006
(25 minutes)
Name:
No books, notes, or calculator allowed. Show detail.
1. 1. (8 points) Consider the following matrix in block form
X=
AB
.
CD
Here A, B , C and D are n n matrices. Assume A is invertible and AC = CA
110.212 Quiz IV Solution
April 7, 2006
(25 minutes)
Name:
No books, notes, or calculator allowed. Show detail.
1. 1. (12 points) Let F[x] be the polynomial algebra over a subeld in C . Find the
monic generator of the following ideals.
(1). Polynomials f (
110.212 Quiz III Solution
March 10, 2006
(25 minutes)
Name:
No books, notes, or calculator allowed. Show detail.
1. (10 points) Let T be a linear operator on a nite dimensional vector space V .
(1). If T 2 = 0. Show that Range(T ) Kernel(T ).
(2). If rank
110.212 Quiz II Solution
February 24, 2006
(25 minutes)
Name:
No books, notes, or calculator allowed.
1. (10 points) Consider three vectors 1 = (1, 0, 0), 1 = (2, 1, 0), 1 = (3, 5, 1) in R3 .
Verify that they form a basis in R3 and write the vector (2, 1,
110.212 Quiz I Solution
February 10, 2006
(25 minutes)
Name:
No books, notes, or calculator allowed.
1. (10 points) Find A1 and write its entries in the form of a + b 2 where a, b Q.
1 1 2
A=
0 2+ 2
Solution: Use row operations on cfw_A | I to get cfw_I
110.212 Midterm Solutions
March 29, 2006
(60 minutes)
No books, notes, or calculator allowed. Show detail.
All matrices and vector spaces are over a subeld F in C.
1. (30 points) True or False. If true, give a short proof (you can cite any previous
exampl
Math 212 Quiz # 4 April 8, 2005
(1) True or false: A nonzero polynomial f (x) over a eld K always has a root, i.e. some
c K such that f (c) = 0. Justify your answer. Solution. False. Just consider
f (x) = x2 + 1 over the real numbers.
(2) Give the denitio
Math 212 Quiz # 3 March 23, 2005
(1) Suppose that T : R3 R3 is a linear map given by the matrix:
1 21
[T ]B = 0 1 1
1 3 4
in the standard basis B . Find a basis for the kernel and range of T . Solution. Find
the reduced row echelon form of [T ]B :
1 0 1
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Math 212 Quiz # 1 February 11, 2005
124
(1) Find the inverse of the matrix: 0 1 2
101
(2) Recall that F2 is the eld with two elements. Determine the number of matrices
A M at34 (F2 ) which satisfy both of the following conditions:
(a) A is in reduced row
Math 212 Quiz # 1 Solutions February 11, 2005
124
(1) Find the inverse of the matrix: 0 1 2 Solution. Find elementary row operations
101
1 2 0
to bring the matrix to reduced row echelon form. The inverse is: 2 3 2
1 2
1
(2) Recall that F2 is the eld with
Math 212 Quiz # 2 February 25, 2005
(1) Let W1 and W2 be subspaces of a vector space V such that W1 + W2 = V and
W1 W2 = cfw_0. Prove that for each vector v V there are unique vectors w1 W1
and w2 W2 such that v = w1 + w2 . Solution. By denition of the su
110.212 Final Exam and Solutions
May 16, 2006
(180 minutes)
All matrices and vector spaces are over a subeld F in complex numbers
C. In some problems, however, we specify the eld as either real numbers R
or complex numbers C.
1. (25 points) Let T : V W be