Special subspaces & linear transformations
Math 4A Scharlemann
15 February 2013
1
CELL PHONES OFF
2
Midterm Friday!
Must bring
TA.RD.IS code (on Gauchospace)
Seat assignment (on Gauchospace - same as
Linear independence
Spanning
Basis of a vector space
Basis & Bases
Math 4A Scharlemann
20 February 2013
1
Linear independence
Spanning
Basis of a vector space
CELL PHONES OFF
Wednesday with the Profes
MATH 4A - ANSWERS FOR MIDTERM 2.A
(1) (a) See Lecture 13 slide 10.
Jack tells Jill that he has a 3 3 matrix A so that
A = A1
Jill says Jacks wrong, there is no such matrix of real
numbers. Who is righ
MATH 4A - ANSWERS FOR MIDTERM 2.B
(1) (a) See Lecture 13 slide 10.
Jack tells Jill that he has a 3 3 matrix A so that
A = A1
Jill says Jacks wrong, there is no such matrix of real
numbers. Who is righ
MATH 4A - ANSWERS FOR MIDTERM 2.C
(1) (a) See Lecture 13 slide 10.
Jack tells Jill that he has a 3 3 matrix A so that
A = A1
Jill says Jacks wrong, there is no such matrix of real
numbers. Who is righ
MATH 4A - ANSWERS FOR MIDTERM 2.D
(1) (a) See Lecture 13 slide 10.
Jack tells Jill that he has a 3 3 matrix A so that
A = A1
Jill says Jacks wrong, there is no such matrix of real
numbers. Who is righ
Describing a vector from a basis
Coordinates of a vector
Changing basis and coordinates in Rn
Vector coordinates with respect to a basis
Math 4A Scharlemann
25 February 2013
1
Describing a vector from
All bases are equal
Dimension, nite and
Dimension of subspaces
dim Nul and dim Col
Dimension and Rank
Math 4A Scharlemann
27 February 2013
1
All bases are equal
Dimension, nite and
Dimension of subs
Eigenstu
Math 4A Scharlemann
1 March 2013
1
CELL PHONES OFF
2
A linear transformation T may be complicated, but look very
simple on some vectors:
Denition
Let T : V V be a linear transformation. Suppo
Finding eigenvalues
Math 4A Scharlemann
4 March 2013
1
CELL PHONES OFF
2
Last time: Find eigenvalues of
4
0
A=
0
0
triangular matrix
1 1 7
5 2 4
02
3
0 0 13
Answer: is an eigenvalue A I4 has non-trivi
More wolves and rabbits
Powers of matrices
Diagonalization
Triangular example
Diagonalization
Math 4A Scharlemann
6 March 2013
Finding P
Finding P 1
More wolves and rabbits
Powers of matrices
Diagonal
Inner (dot) product
Math 4A Scharlemann
8 March 2013
1
CELL PHONES OFF
2
v1
u1
v2
u2
Suppose u , v Rn are given by u = . , v = .
.
.
.
.
vn
un
Denition
The inner product (also called dot product) o
Orthogonal sets
Math 4A Scharlemann
11 March 2013
1
CELL PHONES OFF
2
Recall: Suppose cfw_u1 , u2 , . . . um V is a set of linearly
independent vectors. Then any linear combination can be written
in
Inconsistent system?
Applying vector thinking
Toy example
General case - the picture
General case - an example
Getting close (least squares)
Math 4A Scharlemann
13 March 2013
1
Inconsistent system?
Ap
Review for Final
Math 4A Scharlemann
15 March 2013
1
CELL PHONES OFF
2
Final: Here 8 a.m. Monday. Do only 10 (ten) out of 12 problems.
Must bring
TA.RD.IS code (on Gauchospace)
Seat assignment (on Gau
MATH 4A - FINAL.A - SOLUTIONS
7
2
5
(1) In A = 3 3 6 note that one column is the sum of the other
6 1 5
two. Find three solutions to Ax = 0.
Answer: From Midterm 1.D problem 1 solution
The rst column
MATH 4A - FINAL.B - SOLUTIONS
1
2
3
(1) In A = 6 1 5 note that one column is the sum of the other
3 2 5
two. Find three solutions to Ax = 0.
Answer: From Midterm 1.C problem 1 solution
The third colum
MATH 4A - FINAL.D - SOLUTIONS
4
1
5
(1) In A = 5 3 2 note that one column is the sum of the other
2 1 3
two. Find three solutions to Ax = 0.
Answer: From Midterm 1.A problem 1 solution
The third colum
Linear Equations
Systems of linear equations
Solving a linear system
Linear Equations
Math 4A Scharlemann
UCSB
7 January 2013
Some examples
Possibilities
Linear Equations
Systems of linear equations
S
Review
Getting to echelon
Echelon examples
Echelon form
Math 4A Scharlemann
9 January 2013
Reduced echelon is cool
Review
Getting to echelon
Echelon examples
CELL PHONES OFF
Reduced echelon is cool
Re
Vectors
Math 4A Scharlemann
11 January 2013
1/1
CELL PHONES OFF
No oce hours today
2/1
An m-vector [column vector, vector in Rm ] is an m 1 matrix:
a1
a2
a = a3
.
.
.
am
Can add two m-vectors in th
Solution sets
Math 4A Scharlemann
16 January 2013
1/1
CELL PHONES OFF
Wednesday with the Professor tonight (see syllabus)
2/1
System of linear equations is a bunch of linear equations:
a11 x1 + a12 x2
Chemistry
Nutrition
Economics
Applications
Math 4A Scharlemann
18 January 2013
1
Chemistry
Nutrition
Economics
CELL PHONES OFF
2
Chemistry
Nutrition
Economics
First application: Balancing chemical Rea
The denition and examples
How to determine (in)dependence
Connection with span and a picture
Linear Independence
Math 4A Scharlemann
23 January 2013
1/21
The denition and examples
How to determine (in
The denition and examples
Pictorial examples
Sample problems
The big theorem
Linear Transformations
Math 4A Scharlemann
25 January 2013
1
The denition and examples
Pictorial examples
Sample problems
T
Review for Midterm 1
Math 4A Scharlemann
28 January 2013
1
CELL PHONES OFF
Review begins when everyone is seated
PLEASE TAKE YOUR
ASSIGNED SEAT
2
PLEASE TAKE YOUR
ASSIGNED SEAT
3
x2 = mx1 + b
Equation
MATH 4A - MIDTERM 1.A - SOLUTIONS
(1) [See problem 1.7.31 or lecture 5]
4
1
5
5 3 2
In A =
2 1 3 note that one column is the sum of the
1
0
1
other two. Find three different solutions to
Ax = 0.
Answ
MATH 4A - MIDTERM 1.B - SOLUTIONS
(1) [See problem 1.7.31 or lecture 5]
3
6
3
6 11 5
In A =
4 11 7 note that one column is the sum of the
3
4
1
other two. Find three different solutions to
Ax = 0.
An
MATH 4A - MIDTERM 1.C - SOLUTIONS
(1) [See problem 1.7.31 or lecture 5]
1
2
3
6 1 5
In A =
3 2 5 note that one column is the sum of the
2
0
2
other two. Find three different solutions to
Ax = 0.
Answ