Math 127 Assignment 1 Due Tues. Sept. 20, 9:30 am in class 1. Solve the following system of linear equations: 2x + 3y - 8z = 1 5x + 7y - 19z = 2 2x + 4y - 9z = 0 2. Find the general solution of following system of linear equations: x1 + 2x2 + x3 + 3x4 + 1
Math 127 Solutions to Mid-term Examination 1 2 1 2 1. (20 points) Given the matrix 2 4 2 3 find a basis of the row space, a basis of the 3 6 3 5 column space, and a basis of the null space. Write the 4-th column of as a linear combination of the previous
Math 127 Sample Final Exam 1. a. Suppose that A is a 7 6 matrix such that the system of equations Ax = 0 has only one solution. Determine the rank of A. Briefly explain. b. Let A be a 5 7 matrix with rank 4. What is the dimension of the null space of A? B
Math 127 Some solutions to Sample Final Exam 1. a. Suppose that A is a 7 6 matrix such that the system of equations Ax = 0 has only one solution. Determine the rank of A. Briefly explain. b. Let A be a 5 7 matrix with rank 4. What is the dimension of the
Math 127 Sample Mid-term Exam
1. Solve the folowing system of equations. Express your answer as a particular solution plus all linear combinations of some fixed vectors. 2x1 + 4x2 + x3 = -2 5x1 + 10x2 + 4x3 - 3x4 = - 5 x1 + 2x2 + x3 - x4 = - 1 2. Find the
Fall 2016
Math127
Solutions to Problems 1
1) The induction beginning is clear: If |X| = 1 then X itself and the empty set are the
only subsets of X and so |P(X)| = 2 = 2|X| .
Induction step: Assume now |X| 2 and that the assertion is known for sets Y
with
Fall 2016
Math 127
Problems 2
09/21/2016
1) Show that the set C = R2 is a field with addition
(a, b) + (c, d) := (a + c, b + d)
and multiplication
(a, b) (c, d) := (ac bd, ad + bc) .
2) Let F be a field. Set for a subset X of F
X := x | x X .
Assume the
Fall 2016
Math 127
Problems 1
09/14/2016
1) Let X be a finite non empty set. Denote by P(X) the set of all subsets
of X. Show by induction on the cardinality |X| of X that
|P(X)| = 2|X| .
(This is also true for the empty set as it has only the empty set a
Fall 2016
Math127
Solutions to Problems 2
1) Straightforward verification of the axioms.
2) We observe first that in every field F we have
(1F ) (1F ) = 1F .
In fact, we have seen in class that 0F 0F = 0F , and therefore
0F = (1F + (1F ) (1F + (1F ) .
We
MATH 127: HONORS LINEAR ALGEBRA I
STEFAN GILLE
1
2
STEFAN GILLE
1. Preliminaries: Naive set theory and the principle of induction
1.1. Sets. A set is a collection of objects.
An object a in a set A is also called an element of this set. For brevity we say
Math 127 Assignment 6 (due October 31)
1. (3) Let V be a vector space and let v V have cv = 0 for some nonzero scalar c. Show
that v = 0, mentioning all of the properties of V that you use.
2. (3 + 3) Let V be a general vector space. Prove or disprove:
a)
Math 127 Assignment 8 (due November 14)
1. (3 + 3) a) Let C be the R-vector space of Example 3.7. Prove that dim RC = 2.
b) Let E1 E2 be 2-dimensional subspaces of R3. Show that E1 E2 = Span(v) for some
non-zero v R3.
2. (6) Let A Mmn(F) and define Row(A)
Math 127 Assignment 7 (due November 7)
2
1. (5) Let v1 = 3 , v2 =
4
1
0 ,v =
3
1
1
2 , v =
4
3
1
1 . Find a basis of Span(v , v , v , v ).
1 2
3
4
1
1 1 0 1 1 1
2. (6) Let V be the subspace of M2(R) with basis B = cfw_
,
,
.
0 1 1 0 1 0
2 1
Det
Math 127 Solutions 6
1. Observe that c 0 implies that c 1c = 1 for some scalar c 1. Thus v = 1v = (c 1c)v =
c 1(cv) = c 1(0) = 0, by properties e), d), the hypothesis cv = 0 scalar multiplied by c 1,
and Proposition 3.10 d), respectively.
2. a) This is tr
Math 127 Solutions to Assignment 9 1. Define the linear transformation T : R3 R3 by T (v) = the projection of v onto the vector w = (1, 2, 1). Find the (standard) matrix of T . Compute T (e1 ), T (e2 ), T (e3 ), put the resulting vectors as columns of the
Math 127 Assignment 9 Due Tues. Dec. 5, 9:30 am 1. Define the linear transformation T : R3 R3 by T (v) = the projection of v onto the vector w = (1, 2, 1). Find the (standard) matrix of T . 2. Let T : R2 R2 be a linear transformation such that T (1, 1) =
Solutions to Math 127 Assignment 8 1. Let A be the matrix -5 -3 18 10
.
Find an invertible matrix X so that X -1 AX is diagonal. Use this to find a square root of the matrix A. det A - xI = (-5 - x)(10 - x) + 54 = x2 - 5x + 4 = (x - 1)(x - 4). For = 1, so
Math 127 Assignment 2 Due Tues. Sept. 27, 9:30 am in class 1. Solve the following system of linear equations. Write the solutions in the form of a vector plus all linear combinations of some other vectors. x1 + 2x2 - x3 + x4 + x5 = 3x1 + 6x2 - 2x3 + 4x4 +
Math 127 Solutions to Assignment 2 1. Solve the following system of linear equations. Write the solutions in the form of a vector plus all linear combinations of some other vectors. x1 + 2x2 - x3 + x4 + x5 2x1 + 4x2 - x3 + 3x4 + 4x5 3x1 + 6x2 - 2x3 + 4x4
Math 127 Assignment 3 Due Tues. Oct. 4, 9:30 am 1. Let V = Spancfw_(1, 2, 3), (2, 3, 4), (3, 4, 5), (4, 5, 6). Delete one or more vectors so that the remaining ones span V and are linearly independent. 2. For each of the following, find three linearly ind
Math 127 Solutions to Assignment 3 1. Let V = Spancfw_(1, 2, 3), (2, 3, 4), (3, 4, 5), (4, 5, 6). Delete one or more vectors so that the remaining ones Span V and are linearly independent. We know that if one of these vectors is a linear combination of th
Math 127 Solutions to Assignment 4 1. Let v1 = (1, 2, 3, 4), v2 = (2, 4, 6, 1). Find v3 , v4 so that cfw_v1 , v2 , v3 , v4 is a basis of R4 . A= 1 2 3 4 2 4 6 1 = 1 2 3 4 0 0 0 -7
One way to do this is to pick two more rows, one with a leading entry in c
Math 127 Assignment 5 Due Thurs. Oct. 20, 9:30 am 1. Consider the matrix A= 1 1 3 2 2 2 6 4 10 2 14 20 e + 2e 2 0 4 2 2 2 - 2 .
Just by looking at the columns, without doing any detailed calculations, find the rank of A, and bases for the row space and c
Math 127 Solutions to Assignment 5 1. Consider the matrix A= 1 1 3 2 2 2 6 4 10 2 14 20 e + 2e 2 0 4 2 2 2 - 2 .
Just by looking at the columns, without doing any detailed calculations, find the rank of A, and bases for the row space and column space of
Math 127 Assignment 6 Due Tues. Nov. 8, 9:30 am 1. Write A = 1 3 2 5 as a product of elementary matrices.
2. Prove the law of cosines: if v and w are vectors in R2 , then |v - w|2 = |v|2 + |w|2 - 2|v|w| cos where is the angle between v and w. 3. Find the
Math 127 Solutions to Assignment 6 1. Write A = 1 3 2 5 as a product of elementary matrices.
Subtract twice the first row from the second, that is, multiply A on the left by E1 = 1 0 , giving -2 1 1 3 E1 A = . 0 -1 Multiply row 2 by -1, that is, multiply
Math 127 Assignment 7 Due Thurs. Nov. 17, 9:30 am 1. Find det A, where A is the matrix 1 3 1 3 -2 -3 1 4 A= 1 -1 0 1 4 -1 4 0 2. Suppose that A is a 3 3 matrix whose determinant is 4. If B = 2A and C = 4B -1 , find det C. 3. Show that a+x b+y c+z a b c de
Math 127 Solutions 8
1. a) cfw_1, i is a basis of C as R-vector space: it spans because every z C can be written
z = x + yi with x and y in R, and is linearly independent since x + yi = 0 implies x2 + y2 =
(x yi)(x + yi) = (x yi)0 = 0 with x, y R so x = y