Math 236 Winter 2017
Due Friday, February 10
1. Define T L(R3 , R2 ) in the standard bases by
T (x, y, z) = (x + y + z, 3x 2z).
(a) Find M (T ) with respect to the standard bases for R3 and R2 .
(b) Find M (T ) with respect to the standard basis for R3 an
MATH 236: Assignment 3
Due date: Friday Feb. 13, 1:35 PM
Unless otherwise specified, all vector spaces in this assignment are assumed to be finitedimensional.
1. (15 points) In each of the following, find a basis for the kernel and the image of f . Hint:
MATH 236: Assignment 2
Due date: Monday Feb. 2, 1:35PM
1. (12 points) Find a basis for the following vector spaces (make sure to prove its a basis).
What is the dimension?
(a) cfw_(x1 , . . . , x4 ) F4 | x1 + 3x2 x4 = x3 + x4 = 0
(b) Mm,n (F), the space o
MATH236: Assignment 1
Due date: Friday Jan. 16, 1:35 PM
Please:
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Homework 1 (Friday, January 6th)
Math 236 Algebra 2
Rebecca Patrias
Art Building W-120
Sophie Jiayi Zhu
(#260632892)
Homework 1 (Friday, January 6th)
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Rebecca Patrias
Art Building W-120
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Math 256 Winter 2017
Name:
Quiz
1. Read the proof of the following theorem and fill in the indicated missing details.
Theorem Suppose V is finite-dimensional and U is a subspace of V . Then there is a subspace
W of V such that V = U W .
a Thus there is a
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Math 256 Winter 2017
Due Friday, January 13
1. Decide whether or not U is a subspace of V in the following.
(a) V = C2 and U = cfw_(x, y) C2 | xy = 0
(b) V = R4 and U = cfw_(x1 , x2 , x3 , x4 ) R4 | x1 x2
(c) V = R2 and U = cfw_(x, cos y) | x, y R
2. (Ax
Math 256 Winter 2017
Name:
Example Question
1. Read the proof of the following theorem and fill in the indicated missing details.
Theorem Suppose U1 , . . . , Um are subspaces of V . Then U1 + + Um is a direct sum if and
only if the only way to write 0 as
Math 256 Winter 2017
Due Friday, February 3
1. Consider the vector space R3 over R. Let B = (1, 0, 1), (0, 1, 1), (1, 1, 0) and C = (4, 3, 3), (1, 2, 1), (3, 1, 5)
be two bases.
(a) Let v = (1, 0, 1), expressed in the standard basis for R3 . Find [v]B and
Math 256 Winter 2017
Due Friday, January 27
1. Find a basis for the following vector spaces. In each case, prove it is a basis and state the
dimension of the vector space.
(a) cfw_(x1 , x2 , x3 , x4 ) F4 | x1 + 3x2 x4 = x3 + x4 = 0
a b
1 2
a b
(b)
M2 (
Math 236 Winter 2017
Due Friday, February 17
1. In this exercise, you will work out the details of the proof of the following lemma from Professor
Gorens notes.
Lemma: Let T be a nilpotent operator on an n-dimensional vector space V . Then T n = 0,
where
Math 236 Winter 2017
Due Friday, March 24
1. (Ax 6.A.8) Suppose u, v V , |u| = |v| = 1, and hu, vi = 1. Prove that u = v.
2. (Axler 6.B.2) Suppose e1 , . . . , em is an orthonormal list of vectors in V . Let v V . Prove that
|v|2 = |hv, e1 i|2 + + |hv, em
Math 256 Winter 2017
Due Friday, January 6
1. (Ax 1.A.7) Show that for every C, there exists a unique C such that + = 0.
2. (Ax 1.A.11) Explain why there does not exist C such that
(2 3i, 5 + 4i, 6 + 7i) = (12 5i, 7 + 22i, 32 9i).
3. (Ax 1.B.4) The empty
Math 236 Winter 2017
Due Monday, March 6
1. Consider a matrix A Mn (Z) such that det(A) = 1. Are the entries of A1 necessarily
integers? Explain your answer.
2. Let v1 = (1, 2, 3), v2 = (1, 0, 1), and v3 = (1, 2, 0) be vectors in R3 . Use determinants to
Math 236 Winter 2017
Due Friday, March 17
1. (Ax 5.B.9) Suppose V is finite-dimensional, T L(V ), and v V with v 6= 0. Let p be a
nonzero polynomial of smallest degree such that p(T )v = 0. Prove that every zero of p is an
eigenvalue of T . (See hint belo
Math 256 Winter 2017
Due Friday, January 20
1. (Ax 2.A.1) Suppose v1 , v2 , v3 , v4 spans V . Prove that the list
v1 v2 , v2 v3 , v3 v4 , v4
also spans V .
2. Find a number t such that
(3, 1, 4), (1, 5, 9), (2, 6, t)
3
is not linearly independent in R . E
Sample Final
All vector spaces are assumed to be finite dimensional, and F always denotes a field.
1. Define the following terms carefully.
(a) an eigenspace attached to an eigenvalue of a linear operator.
(b) a linearly independent set of vectors.
(c) th