Math 22 Review for Midterm 2
Just as for the rst midterm, the exam will contain questions much like those from the
homework. If youve been doing well there and understand your answers and the concepts
behind them, youll do well.
Vector Space Terminology/C
Math 22, Exam II
May 13, 2010
NAME:
This is a closed book exam and you may not use a calculator. Use the
space provided to answer the questions and if you need more space, please
use the back of the exam making sure to write a note in the space provided
t
Ungraded Quiz + Questionnaire 3
Your name:
May 3, 2014
1. Let V = P2 and let B = cfw_1, t, t2 be the standard basis for P2 . Compute [1 4t + 7t2 ]B .
If 1 4t + 7t2 = c1 (1) + c2 (t) + c3 (t2 ) then
c1
1
c2 = 4 .
c3
7
This is the B-coordinate vector [1
Ungraded Quiz + Questionnaire 3
Your name:
April 25, 2014
1. Is cfw_1, t a basis for the vector space P2 of all polynomials with degree 2?
NO. Any basis for P2 has three elements, because the basis cfw_1, t, t2 has three elements. Thus cfw_1, t is
not a
Ungraded Quiz 2 + Questionnaire 2
Your name:
April 7, 2014
1. Is the system
2000293x + 9323909y + 2014z = 0
323123x
407y + 2187z = 0
consistent?
Yes. It is a homogeneous system, and homogeneous systems are always consistent (the zero vector is
a solution
Lecture 4: Ax = b and solution sets
Danny W. Crytser
March 28, 2014
Todays lecture
We saw in the previous lecture that solving systems of linear
equations is equivalent to solving certain vector equations
x1 a1 + x2 a2 + . . . + xp ap = b ().
Dan Crytser
Math 22, Exam I
April 22, 2010
NAME:
This is a closed book exam and you may not use a calculator. Use the
space provided to answer the questions and if you need more space, please
use the back of the exam making sure to write a note in the space provided
Lecture 8: The matrix of a linear transformation.
Applications
Danny W. Crytser
April 7, 2014
Example
Let T : R2 R3 be the linear transformation dened by
x1 + x2
x1
T
= 2x1 .
x2
3x2
Dan Crytser
Lecture 8: The matrix of a linear transformation. Application
Lecture 6: Linear independence
Danny W. Crytser
April 2, 2014
Todays lecture
1
Suppose we have vectors a1 , . . . , ap in Rn .
Dan Crytser
Lecture 6: Linear independence
Todays lecture
1
Suppose we have vectors a1 , . . . , ap in Rn . When does the
homoge
Math 22: Linear Algebra
Danny W. Crytser
Dartmouth College
March 24, 2014
What is a linear equation?
Dan Crytser
Math 22: Linear Algebra
What is a linear equation?
A linear equation is an equation of the form
a1 x1 + . . . + an xn = c
where the a1 , . . .
Lecture 7: Linear transformations
Danny W. Crytser
April 4, 2014
Todays lecture
1
We will review the concepts of sets and functions
Dan Crytser
Lecture 7: Linear transformations
Todays lecture
1
We will review the concepts of sets and functions
2
We will
Lecture 3: Vector equations
Danny W. Crytser
March 26, 2014
Todays lecture
We have seen that the solution sets to linear equations can often
be described as lines in the plane.
Dan Crytser
Lecture 3: Vector equations
Todays lecture
We have seen that the s
Ungraded Quiz + Questionnaire
Math 22: 12 Section
March 28, 2014
1. Solve the linear system
x+ y =1
x + 2y = 1
Solution:
Write the augmented matrix
1
1
1
2
1
1
.
Subtract the rst from the second row, transforming the matrix to echelon form
1
0
1
1
1
0
.
T
MATH 22 : Linear Algebra with Applications
Naomi Tanabe
Department of Mathematics
Dartmouth College
Nov 7 - Nov 11, 2016
Least-Squares Problems
Problem (1)
4 0
Solve the matrix equation Ax = b where A = 0 2 and
1 1
2
b = 0 .
11
Definition
Let A be an m
MATH 22 : Linear Algebra with Applications
Naomi Tanabe
Department of Mathematics
Dartmouth College
Oct 10 - Oct 14, 2016
Basis
Recall that an indexed set of vectors cfw_v 1 , . . . , v r is said to be
linearly independent if and only if the vector equat
MATH 22 : Linear Algebra with Applications
Naomi Tanabe
Department of Mathematics
Dartmouth College
Oct 3 - Oct 7, 2016
Determinants
Definition
For n 2, the (i, j)-cofactor of an n n-matrix A = [aij ] is the
number given by
Cij = (1)i+j det(Aij ).
Definit
MATH 22 : Linear Algebra with Applications
Naomi Tanabe
Department of Mathematics
Dartmouth College
Sept 19 - Sept 23, 2016
Matrix Equations
Theorem
Let A be an m n-matrix with columns ai , i.e., A = [a1 an ]
and b Rn . Then the matrix equation
Ax = b
and
MATH 22 : Linear Algebra with Applications
Naomi Tanabe
Department of Mathematics
Dartmouth College
Oct 24 - Oct 28, 2016
Diagonalization
Definition
A square matrix A is said to be diagonalizable if A = PDP 1
for some invertible matrix P and diagonal matr