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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make sure your assignment is securely stapled together;
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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
Sophie Jiayi Zhu
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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
Math 256 Winter 2017
Name:
Quiz
1. Consider the following list of vectors in R3 (considered as a vector space over R).
(1, 0, 1), (1, 2, 3), (1, 0, 0), (0, 1, 1)
(a) Without using any computations, explain how you know this list is not linearly independen
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 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