Summary of Day 21
William Gunther
June 19, 2014
1
Objectives
Look at projections between two vectors, and generalize to projection of a vector
on a space.
2
Summary
In another class you might have explored the idea of a projection of one vector onto
ano
Homework 2 Solutions
2.3.21 (a) Suppose that w is a linear combination of vectors u1 , . . . , uk and that each ui is a linear combination of vectors v1 , . . . , vm . Prove that w is a linear combination of v1 , . . . , vm .
Proof. Consider ui . We know
Exam 2 Review
June 13, 2014
1
Technique Question
The following is a list of question. Contained in this list are some that will be chosen to put on the test.
Note that none of these proofs needs to be particularly long. Most are about a paragraph in lengt
Exam 3 Review
June 24, 2014
1
Technique Questions
1. Let W be a subspace of Rn .
Prove that W is a subspace of Rn .
If v1 , . . . , vm is a basis for W and u1 , . . . , uk is a basis for W then v1 , . . . , vm , u1 , . . . , uk is a
basis for Rn .
(Note
Exam 1 Review
May 29, 2014
1
Technique Question
The following is a list of question. Contained in this list are some that will be chosen to put on the test.
Note that none of these proofs needs to be particularly long. Some can be done with two sentences.
Homework 7 Solutions
5.1.36 If n > m then there is no m n matrix A such that |Ax| = |x|. (Hint: this has not much to do with
norms).
Solution. By the rank-nullity theorem, the nullity must be larger than 0. x null(A) where x = 0.
Then |x| = 0 but |Ax| = |
Homework 3 Solutions
3.5.2 Determine is the set of all (x, y) R2 | x 0 and y 0 is a subspace of R2
Solution. Not a subspace.
1
1
S but
1
1
=
1
. Therefore S is not closed under scalar
1
multiplication.
x
3.5.6 Determine if y R3 such that z = 2x and y
Homework 3 Solutions
3.1.19 A factory produced three products and ships them to two warehouses for storage. The number of
units of each product shipped to each warehouse is given by the matrix:
200 75
A = 150 100
100 125
Where aij is the number of units o
Homework 5 Solutions
3.6.7 Give a counterexample to show that the given transformation is not a linear transformation:
T
x
y
=
y
x2
T
0
1
=
0
1
T
0
2
=
0
4
Solution. Note:
So:
T
0
+T
1
0
2
0
5
=
But
T
0
0
+
1
2
=T
0
3
=
0
9
3.6.44 Let T : R3 R3 be a linea
Homework 6 Solutions
1 Calculate det(A) if:
1
1
A=
2
0
1
3
0
2
3
7
0
2
11
21
4
6
Solution. Do some row reductions (which all involve adding a row to a constant multiple of another
row), and you arrive at the matrix:
1 1 3 11
0 2 4 10
0 0 2 8
0 0 0 4
As
Homework 1 Solutions
2.2.1-8 We will determine if the matrix is in row echelon form, and if so decide whether its in reduced row
echelon form. Note: No work or explanation was required for these problems, but Ill explain what
conditions they violate when
Summary of Day 5
William Gunther
May 23, 2014
1
Objectives
Solidify understanding of the idea of span and linear sombinations.
Dene the notion of linear independence and linear dependence.
Prove some theorems involving linear dependence/independence.
2
Summary of Day 22
William Gunther
June 20, 2014
1
Objectives
Explore Abstract Vector Spaces
2
Summary
This semester we have explored the vector space Rn and subspaces of Rn . We now
move on to more abstract vector spaces whose geometric nature is either
Summary of Day 2
William Gunther
May 20, 2014
1
Objectives
Dene row echelon form (ref), free variable, leading variable, and rank.
Identify elementary row operations and their purpose.
Perform elementary row operations to reduce a matrix to ref (Gaussi
Summary of Day 8
William Gunther
May 29, 2014
1
Objectives
Explore the algebraic structure of inverses.
2
Summary
Today we will be focusing on the following algebraic property of a matrix: an inverse. Let A be a n n
matrix. B is an inverse of A if:
BA =
Summary of Day 7
William Gunther
May 28, 2014
1
Objectives
Connect matrix multiplication with systems of equations.
See the true nature of matrices as functions.
Dene some more matrix operations (e.g. transpose) and explore algebraic properties of matr
Summary of Day 9
William Gunther
May 30, 2014
1
Objectives
Prove the fundamental theorem of inverses.
Talk about subspaces of Rn .
Prove able special subspaces from a matrix.
Dene dimension and basis.
2
Summary
Yesterday we stated this, but did not p
Summary of Day 1
William Gunther
May 19, 2014
1
Objectives
Recognize linear equations.
Dene a system of linear equations.
Build geometric intuition for a solution for a system of linear equations (in R2 and R3 anyway).
Solve a system back back substit
Summary of Day 3
William Gunther
May 21, 2014
1
Objectives
Use Gauss-Jordan elimination to solve a system of linear equations.
Restate problem of system of equations into a problem of vector equations.
Explore the geometry and algebraic properties of v
Summary of Day 6
William Gunther
May 27, 2014
1
Objectives
Talk about relationship between homogeneous equations and linear independence/dependence and prove
some theorems.
Discover another way to approach linear independence with row vectors in a matri
Summary of Day 4
William Gunther
May 22, 2014
1
Objectives
Explore more geometric properties of Rn by looking at dot products to capture the notions of lengths
and angles.
Calculate dot products and norms of vectors.
Write parametric and normal equatio
M241: Matrices and Linear Transformations
Homework Assignment 10
Answers
Graded Problems: Problem A (graded as ve problems), C (2); Bonus.
Section 5.2
Problem 11. (a) It is assumed that A is a 3 3 matrix with eigenvalues 1, 1, 2. Since 0 is not an
eigenva
M241: Matrices and Linear Transformations
Homework Assignment 9
Answers
Graded Problems: Section 5.2 Problem 32; Section 5.3 Problem 24; Problem A (graded as
four problems), Review (graded as bonus).
Section 5.1
Problem 14. Note that rank(A) = 1, this mea
Matrices and Linear Transformations
Homework Assignment 5
Due on Friday, October 11, at the start of class.
Your homework should have the cover sheet posted on Blackboard. If you use several
sheets, please staple them. The problems should be written neatl
Matrices and Linear Transformations
Homework Assignment 6
Due on Thursday, October 17, at the start of your recitation.
Your homework should have the cover sheet posted on Blackboard. If you use several
sheets, please staple them. The problems should be w
Matrices and Linear Transformations
Homework Assignment 4
Due on Friday, October 4, at the start of class.
Your homework should have the cover sheet posted on Blackboard. If you use several
sheets, please staple them. The problems should be written neatly
Matrices and Linear Transformations
Homework Assignment 3
Due on Friday, September 20, at the start of class.
Your homework should have the cover sheet posted on Blackboard. If you use several
sheets, please staple them. The problems should be written nea
Matrices and Linear Transformations
Homework Assignment 1
Due on Friday, September 6, at the start of class.
Your homework should have the cover sheet posted on Blackboard. If you use several
sheets, please staple them. The problems should be written neat
Matrices and Linear Transformations
Homework Assignment 2
Due on Friday, September 13, at the start of class.
Your homework should have the cover sheet posted on Blackboard. If you use several
sheets, please staple them. The problems should be written nea