SOLUTION TO ASSIGNMENT 1 - MATH 251, WINTER 2007
1. Consider the following vector spaces (you do not need to prove those are vector spaces, unless specified): (1) The vector space V1 of continuous functions f : [0, 1] R, where we define f + g to be
McGill University
MATH 251: Algebra 2
Assignment 5 Solutions
1. (a) Since a(u, 0) + b(v, 0) = (au + bv, 0) and a(0, u) + b(0, v) = (0, au + bv) we see that U1 , V1 are subspaces
since (0, 0) U1 , V1 . It also follows that the mappings f1 : U U1 , f2 : V V
McGill University
MATH 251: Algebra 2
Assignment 6 Solutions
1. (a) Since A2 = nA, we have A(A nI) = 0 so that the possible eigenvalues of A are 0, n.
If the characteristic of F does not divide n, the minimal polynomial of A is X 1 if n = 1 and X(X n) is
Least Squares and the Generalized Inverse
An important problem is to nd a polynomial curve y = f (x) which best ts a given set of m
data points (x1 , y1 ), (x2 , y2 ), . . . , (xm , ym ). If f (x) = a0 + a1 x + + an1 xn1 then the given points
lie on the c
The Real Jordan Canonical Form and the Rational Canonical Form
Not all matrices over a given eld have a Jordan canonical form as not all polynomials split
completely into linear factors. For example, over the reals one can have irreducible quadratic facto
McGill University
MATH 251: Algebra 2
Assignment 4 Solutions
1. The function y = f (xf is a solution of the given dierential equation f Ker(D4 6D3 + 13D2 12D + 4),
where D is the dierentiation operator on the vector space of innitely dierentiable real-val
McGill University
MATH 251: Algebra 2
Assignment 3 Solutions
1. (a) If aex + be2x + ce3x = 0 for all x then on dierentiating twice we get
aex + be2x + ce3x = 0
aex + 2be2x + 3ce3x = 0
aex + 4be2x + 9ce3x = 0
for all x. Setting x = 0, we get
a+b+c=0
a + 2b
Notes on the Dual Space
Let V be a vector space over a eld F . The dual space of V is the vector space V = Lin(V, F ) consisting
of the linear mappings : V F . The elements of V are called linear forms or linear functionals. If V has
basis e = (ei )iI the
McGill University
MATH 251: Algebra 2
Solutions to Assignment 8
1. Since P 2 P = 0 we have V = Ker(P2 P) = Ker(P(P1) = Ker(P)Ker(P1) = Ker(P)Ker(Q).
Now v Ker(Q) = v = P(v) = v Im(P) and v Im(P) = v = P(u) = Q(v) =
(1 P)P(u) = 0 = v Ker(Q). Hence Ker(Q) =
MATH 251, Winter 2009
Honours Algebra 2
Solutions to the Midterm Exam
1. Let V = R5 , a real vector space, and W1 , W2 be
of V .
80 1 0
10
3
>1
>
>B C B
>2
<B C B 7 C B
CB
W1 = Span B 0 C , B 2 C , B
CB
>B C B
>@ 0 A @ 1 A @
>
>
:
0
1
80 1 0
10
8
>4
>
>B
SOLUTION TO ASSIGNMENT 2 - MATH 251, WINTER 2007
1. Let B = {(1, 1), (1, 5)} and C = {(2, 1), (1, -1)} be bases of R2 . Find the change of basis matrices B MC 8 and C MB between the bases B and C. Let v = ( 28 ) with respect to the standard basis. F
SOLUTION TO ASSIGNMENT 3 - MATH 251, WINTER 2007
1. Let T : V V be a nilpotent linear operator. Prove that if n = dim(V ) then T n 0. Show that for every n 2 there exists a vector space V of dimension n and a nilpotent linear operator T : V V su
SOLUTION TO ASSIGNMENT 4 - MATH 251, WINTER 2007
1. Deduce from the theorems on determinants the following: (1) If a column is zero, the determinant is zero. (2) det(A) = det(At ), where At is the transposed matrix. (3) If a row is zero, the determi
SOLUTION TO ASSIGNMENT 5 - MATH 251, WINTER 2007
1. (A) Let W be a k-dimensional subspace of Fn . Prove that there are n - k linear equations such that W is the solutions to that homogenous system. (B) Let W1 = Span ({(1, 1, 0, 0), (0, 1, 1, -1)}) a
SOLUTION TO ASSIGNMENT 10 - MATH 251, WINTER 2007
1. Let A be a matrix in block form: A1 0 A= 0 Prove that A = A 1 A 2 A r , and mA = lcm{mA1 , mA2 , , mAr }. You may use the formula Ab 1 0 Ab = 0 for every positive integer b. Proof. We
SOLUTION TO ASSIGNMENT 11 - MATH 251, WINTER 2007
1. It is known that a differentiable function f : R2 R has a maximum at a point P if f /x = f /y = 0 at P and the 2 2 symmetric matrix -
2f x2 2f xy 2f xy 2f y 2
is positive definite; minimum at
SOLUTION TO ASSIGNMENT 9 - MATH 251, WINTER 2007
1. Calculate the characteristic and minimal polynomial of the following matrices with real entries. In each case determine the algebraic and geometric multiplicity of each eigenvalue. Decide which mat
SOLUTION TO ASSIGNMENT 8 - MATH 251, WINTER 2007
In this assignment F = R or C. 1. Consider results of an experiment given by a series of points: (x1 , y1 ), (x2 , y2 ), . . . , (xn , yn ), where x1 < x2 < < xn and the xi , yi are real numbers. W
SOLUTION TO ASSIGNMENT 6 - MATH 251, WINTER 2007
1. Let T : V W be a linear map and define T : W V by (T (g)(v) := g(T v). Prove the following lemma: LEMMA 1. (1) T is a well-defined linear map. (2) Let B, C be bases to V, W , respectively. L
189-251B: Honors Algebra 2
Midterm Exam: Corrections and
comments
The fol lowing was the grade dsitribution in this midterm:
>= 100 5 95 99 5 90 94 13
80 89 13 70 79 6 60 69 7
50 59 1
So overal l the grades ended up being on the high side, and the above w
McGill University
MATH 251: Algebra 2
Assignment 1 Solutions
1. If V is a vector space and f is an isomorphism then f (u + v) = f (u) + f (v), f (cv) = cf (v). Setting
u = f (u), v = f (v), we see that
u + v = f (f 1 (u ) + f 1 (v ),
cv = f (cf 1 (v )
so