Math 140A Suggested Syllabus
Text: Elementary Analysis: The Theory of Calculus, Kenneth A. Ross
Lecture Section Topic
1 The Set of Natural Numbers
2 The Set of Rational Numbers
3 The Set of Real Numbers
4 The Completeness Axiom
10. Basis for Col A:
A72 Answers to Even-Numbered Exercises
dimCol A = 4
Basis for Nul A: l
; dim Nul A = 1
. _ 1 -
Basts ofColA. , 6 . _9 .
dimColA : 3
Extra Questions for Exam Review
1. Find the truth table for the logical expression
( P ^ Q) =) ( R _ P).
2. Are the sets
A = cfw_2a + 1 : a 2 Z
B = cfw_ a : 2a + 1 2 Z
(a) Let A = cfw_2, 4, 6, 8, 10 and B = cfw_1, 4, 7, 10. What i
Student ID #:
Sample Final, Page 2 of 9
a) Write down the definition of conditional probability:
P (E|F ) =
b) Prove that for any three events E, F, G with P (F ) 6= 0:
P (E|F ) = P (EG|F ) + P (E
Extra Practice Problems for the Final
1.) Laplace transform of t sin(8t).
2.) Laplace transform of t2 e3t .
3.) Laplace transform of sin(t)u(t 5).
4.) Inverse Laplace transform of
s2 +4s+9 .
5.) Use Laplace
transforms to solve x00 + 6x0 + 9x = f (t), wh
Quiz 6, Version A MATH 3A, LINEAR ALGEBRA, 831 2017
-1 0 3
LetA= -3 2 3 .
0 O 2
(a) Find the eigenvalues and eigenveetors of A.
(b) Find an invertible matrix P and a diagonal matrix D such that A 1: PDP".
(3) Test the similarity relation A =
Syllabus: Math 3A, Intro to Linear Algebra, class 44230 (Lec A), SS1, 2017
Text: "Linear Algebra and its Applications" (5th Edition) by David C. Lay
Lectures: Mon, Wed, Fri, 1pm - 2:50pm, room SSL 290
Instructor: Vladimir Goren, [email protected]
Math 3A, Intro to Linear Algebra, Class 44230, Summer Session I, 2017
Textbook: "Linear Algebra and its Applications" (5th Edition) by David C. Lay
Subspaces of Rn
The xy -plane is like a copy of hR2isitting
h 1ini R . h 0 i
Every vector in it is of the form y = x 0 + y 1 .
So everything inh the
can be written uniquely as a linear
combination of 0 and 1 .
The sum or
Section 1.9: The Matrix of a Linear Transformation
Is every linear transformation a matrix transformation?
Key idea: Linearity lets us find a lot of images just by knowing a few.
Simplest case: A linear transformation T : R R is completely
Math 3A Spring 2016
Instructor: Ryan Broderick
Office Hours: M 2-2:50, W 4-4:50, and F 1:30-2:30
Weekly quizzes and HW
Your grade will be computed as follows:
Linear algebra is an important and beautiful subj
be any 2 2 matrix. The image of the unit square is
the parallelogram spanned by a~1 = [ ca ] and a~2 = db .
Adding a multiple of one column to another doesnt change the
determinant. Want to show it doesnt change area either.
3.3: Determinants a
Section 1.4: The Equation A~x = ~b
The matrix-vector product
Linear combination x1 a~1 + + xn a~n requires two pieces of
information: the vectors and the weights.
Arrange weights xi into a vector ~x = .
The matrix-vector product
Section 2.1: Matrix Operations
It will be useful to introduce operations on the set of matrices. First, some
notation and terminology.
We use two subscripts to denote the entries of a matrix A: aij is the entry
in the ith row and jth column.
a11 a12 . . .
Recall we classified all the subspaces of R3 geometrically:
Line through ~0
Plane containing ~0
A single nonzero vector
Two linearly independent vectors
Three linearly independent vectors
A line in R2 looks l
5.1: Eigenvectors and Eigenvalues
A linear transformation T can be described by its standard matrix A
This is because A encodes information about where T sends e~1 , . . . , e~n
But sometimes its more convenient to describe T by where it sends
Section 1.8: Linear Transformations
T ([ yx ]) = x y is a transformation from R2 to R. In this case the
range and codomain are both R.
In calculus, you studied functions that take a real number as input
and produce a real number as output.
Section 2.2: Inverse of a Matrix
Solving ax = b (a 6= 0) is very easy: Just divide by a.
Can we do something similar with A~x = ~b?
When we divide by a were multiplying by a1 , a number such that
a1 a = 1, so that a1 ax = 1x = x.
So we want to find a matr
Section 1.7: Linear Independence
4 ]. It consists of all vectors of the form
Consider Spancfw_[ 13 ] , [ 12
c1 + 4c2
c1 + 4c2
= (c1 + 4c2 )
3c1 + 12c2
3(c1 + 4c2 )
So every element of this span is a multiple of
We want a definition of det A for n n matrices that has these same
properties and also satisfies det In = 1.
It turns out there is exactly one function that has all these properties.
To define it, we need to work with submatrices.
3.1: Introduction to Det
What if A doesnt have n distinct eigenvalues? Then well need to find
multiple linearly independent eigenvectors from the same eigenspace.
= (2 )2
2 is the only eigenvalue.
5.4: Eigenvectors and Linear Transformations
Diagonalizing a matrix A means finding D (diagonal) and P (invertible) so
that A = PDP 1 .
Lets interpret this as a composition of linear transformations.
So how can we view A = PDP 1 as a composition of linear
5.2: The Characteristic Equation
We saw that is an eigenvalue of A if and only if A In is singular.
This happens exactly when det(A In ) = 0.
Find the eigenvalues of A =
The equation det(A In ) = 0 is called the characteristic equation
Basis of a subspace
The rest of this section is devoted to finding efficient ways of writing a
subspace as a span.
If H is a subspace of Rn and v~1 , . . . , v~k are vectors in Rn , then we say
cfw_v~1 , . . . , v~k is a spanning set for H if
A solution to a linear system is simply a finite list of numbers. It will
be useful to be able to perform algebraic operations on these which
brings us to the notion of a vector.
Section 1.3: Vector Equations
A matrix with only one column is
and P =
You can check that A = PDP . Find D and A .
Let A =
We saw that if A and B are similar, they have the same eigenvalues.
Can learn a lot about A by studying B
Goal: Find a very sim
Find an elegant way to describe solution sets of linear systems
Visualize solution sets geometrically in the 2 and
We begin with a class of simpler systems.
Section 1.5: Solution sets of linear systems
Section 2.3: Characterizations of Invertibility
So to show A is invertible, its enough to find B with AB = In .
So A1 exists and thus B = A1 AB = A1 In = A1 .
Invertible Matrix Theorem
Similarly if we find a B with BA = In .
Let A be an n n matrix. Then t
3.2: Properties of Determinants
Effect of row operations on the determinant
Let A and B be n n matrices.
If B is obtained from A by rescaling a row by k, then det B = k det A
If B is obtained from A by swapping two rows, det B = det A
If B i