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 last time)
Convincing photo id
8.5 by 11 bluebook for w
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 Professor tonight (see syllabus)
2
Linear independence
Spanni
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 right and why? (Hint: Find det A.)
Answer: Jill is right: t
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 right and why? (Hint: Find det A.)
Answer: Jill is right: t
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 right and why? (Hint: Find det A.)
Answer: Jill is right: t
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 right and why? (Hint: Find det A.)
Answer: Jill is right: t
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 a basis
Coordinates of a vector
Changing basis and coo
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 subspaces
dim Nul and dim Col
CELL PHONES OFF
Wednesday wit
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. Suppose for some
R and v = 0 V,
T (v ) = v
Then v is an eig
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-trivial nullspace
A I4 has free variable is a diagonal entr
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
Diagonalization
Triangular example
Finding P
Finding P 1
CELL P
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) of u and v is the real
number:
u v = u1 v1 + u2 v2 + . .
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 exactly one way. In other words:
c 1 u1 +c 2 u2 +. . .+
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?
Applying vector thinking
Toy example
General case - the p
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 Gauchospace - same as last time)
Convincing photo id
8.5 b
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 is the sum of the last two or, thinking of them
as colu
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 column is the sum of the rst two or, thinking of them
as col
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 column is the sum of the rst two or, thinking of them
as col
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
Solving a linear system
CELL PHONES OFF
Some examples
Po
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
Review
Getting to echelon
Echelon examples
Reduced echelo
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 the obvious way, or multiply a vector by
a real number:
a
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 + . . . + a1n xn = b1
a21 x1 + a22 x2 + . . . + a2n xn
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 Reactions
The chemicals hydrazine (N2 H4 ) and nitrogen te
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)dependence
Connection with span and a picture
CELL PHO
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
The big theorem
CELL PHONES OFF
2
The denition and examp
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 of a line:
x1
0
1
=
+
x
x2
b
m1
x2
This is sum of spec
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.
Answer: The third column is the sum of the rst two or, thin
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.
Answer: The second column is the sum of the other two or,
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.
Answer: The third column is the sum of the rst two or, thin