Definitions
2.3
• If S = {v1, v2, …, vk } is a set of vectors in Rn, then the set of all linear combinations of v1,
v2, …, vk is called the span of v1, v2, …, vk and is denoted by span(v1, v2, …, vk) or
span(S). If span(S) = Rn, then S is called a spannin
Name:
Linear Algebra - Math215003
Quiz 2 9/13/10
As always, in order to receive full credit7 you must show ALL of your workl You have NB minutes to
complete the following problems.
1. Given
2 6
A = 7 21
*3 9
Find the parametric vector form of the solutio
Linear Algebra - Math215-003 Weekend Challenge Quiz 1 - 9/1/10 - Name: As always, in order to receive full credit, you must show ALL of your work! On Weekend Challenge Quizzes (WCQs) you may use your notes, the text, and discuss problems with your classma
Name: Linear Algebra - Math215-003 Weekend Challenge Quiz 2 - 9/10/10 As always, in order to receive full credit, you must show ALL of your work! On Weekend Challenge Quizzes (WCQs) you may use your notes, the text, and discuss problems with your classmat
MATH 215: CHAPTER 7 SOLUTIONS
7.2 #5. If u = c1 v1 + + cn vn and w = d1 v1 + + dn vn , then in the notation of the
problem,
c1
d1
.,
.
[w] = . .
[u] = .
.
.
dn
cn
Then
u, w
=
c1 v1 + + cn vn , d1 v1 + + dn vn
n
=
ci vi , d1 v1 + dn vn
i=1
n
n
=
ci dj
MATH 215, SOLUTIONS FROM SEC 5.3
#2. Let U be the set of symmetric n-by-n matrices, and W the set of skewsymmetric matrices. This problem asks you do a number of things.
Prove that U , W are vector spaces: Well give a detailed proof that U is a
vector spa
MATH 215: SECTION 5.1 HOMEWORK SOLUTIONS
#1.
(a) To show that three vectors form a bases of R3 , it suces to show that they are linearly
independent. So, we need the matrix
4 5 1
A = 2 2 3
1 3 0
to have three pivots in row echelon form; equivalently, we
MATH 215: SECTION 6.2 HOMEWORK SOLUTIONS
#1.
(a) T1 is not a linear transformation. To show this, all we have to do is show that
the rules for linear transformations are violated. For instance, T1 ([1 0 0]) = 1 and
T1 ([0 1 0]) = 1, but
T1 ([1 1 0]) = 2 =
MATH 215: SECTION 6.3 HOMEWORK SOLUTIONS
#1.
(a) This linear transformation is given by the matrix 1 1 1 1 : this is because
x1
x2
1 1 1 1 = (x1 + x2 + x3 + x4 ) .
x3
x4
By the usual procedure, we can verify that a basis for the nullspace of 1 1 1 1
i
MATH 215: CHAPTER 4 HOMEWORK SOLUTIONS
4.1 #2.
(a) The set of all column vectors in R3 of length 1 is not a vector space, for many reasons.
A vector space must contain the zero vector, but the zero vector does not have length
1. The scalar product of c R
ACCON lC
Linear Algebra l\/lath215003 Quiz 1 - 9/1/10 - Name:
As always, in order to receive full credit, you must Show ALL of your work!
You have ~5 minutes to complete the following problems.
1. (4pts) Do the following three lines have a common point of
Name:
Linear Algebra - Math215-003
Weekend Challenge Quiz 6 - Due 11/1/10
BE SURE TO FOLLOW THE SUBMISSION GUIDELINES ON THE WEBSITE!
As always, in order to receive full credit, you must show ALL of your work!
Consider the vector spaces SY M 2 - the space
Definitions 3.1 A matrix is a rectangular array of number called the entries, or elements, of the matrix. If A is an m x n matrix and B is an n x r matrix, then the product C = AB is an m x r matrix. The ( i, j ) entry of the product is computed as f
Practice for Test 1 For full credit, show all work. 1 State the precise definition of the following concepts: (a) span of a set of vectors in Rn . (b) Linear combination. 1 1 0 2 Let u = 2 , v = 1 , w = 1 3 0 1 (a) Compute the distance an
Practice Test 3 For full credit, show all work. You can use your calculator for computational purposes. But your work on paper must be transparent enough that I understand your answer without a calculator. No credit will be rewarded if I cannot under
Practice Test 2 For full credit, show all work. 1 State the definition for the following: a) A basis for a subspace S. b) An invertible matrix M . c) rank(A). 3 1 4 6 2 Given A = 2 0 1 3. Find the following: -1 1 2 0 a) A basis for null(A). b) A
Name:
Linear Algebra - Math215-003
Optional Weekend Challenge 1 - Due 10/4/10
BE SURE TO FOLLOW THE SUBMISSION GUIDELINES ON THE WEBSITE!
As always, in order to receive full credit, you must show ALL of your work!
This will be graded and count as an extra
ACCON lC
Linear Algebra l\/lath215003 Quiz 1 - 9/1/10 - Name:
As always, in order to receive full credit, you must show ALL of your work!
You have ~5 minutes to complete the following problems.
1. (4pts) Do the following three lines have a common point of
Name:
Linear Algebra - Math215003
Quiz 2 9/13/10
As always, in order to receive full credit7 you must show ALL of your workl You have NB minutes to
complete the following problems.
1. Given
2 6
A = 7 21
*3 9
Find the parametric vector form of the solutio
Name: b!
Linear Algebra - Math215-003
Quiz 3 - 9/29/10
As always, in order to receive full credit, you must show ALL of your work! You have ~10 minutes
to complete the following problems.
1. (9pts) Given
A=[_13§], kl
Find the following matrixroduct