The parts of the interpretations not specied can be dened arbitrarily.
A. 10. (x)(y )[P x Qy (z )(Rxy Sxyz )];
S = U, P, Q, R, S
true: U = cfw_a
false: U = cfw_a
11. (x)(y )[P x Ry Sxy (z )(P x Rx Qxaz )];
true: U = cfw_a
Solutions to Review Problems for Exam 1.
7. (H S ) P or P (H S )
8. H S P or (H S ) P
11. (S H ) P
12. O V (G L) or O V G L
13. N (M L)
14. C R P
3. Not Valid.
7. Not Valid.
Solution for Exam 1
1. a) Solve the following initial-value problem:
+ 3ty = 2t,
y (0) = 1.
This equation is linear (and separable), with integrating factor
I = exp
= exp(3t2 /2).
Therefore, we get
(Iy ) = 2t exp(3t2 /2) Iy
MA 262 PRACTICE PROBLEMS
Circle the letter corresponding to your choice of correct answer.
1. Find the determinant of A if
2. If A is a 3 4 matrix and P = 2 and Q = 2 are
Study Guide for MA 262
Linear Algebra and Dierential Equations
Topics covered in the exam
1. Complex numbers, polar representation, roots of complex numbers.
2. Vector spaces: subspace, basis, spanning set, linear combination, linear indep
Solutions to Section 1.1
1. FALSE. A derivative must involve some derivative of the function y = f (x), not necessarily the rst
2. TRUE. The initial conditions accompanying a dierential equation consist of the values of y,
main 2007/2/16 page 189 i
Mathematics is the gate and key to the sciences. Roger Bacon
In this chapter, we introduce a basic tool in applied mathematics, namely the determinant of a square matrix. The determinant is a number,
Systems of Linear Equations
Consider the general systems of equations
Is there a solution?
If so, how many?
How do we nd solutions?
What does this look like in the plane for the following:
A solution corresponds to the coordinates of an i
This technique will be used for DEs in dierential form
M(x,y)dx + N(x,y)dy = 0
Any function phi satisfying these properties is called a potential function
for Mdx + Ndy = 0
What's the point?
If a DE is exact and we can nd a potential function p
2.1 Matrices denitions and notation
2.2 Matrix Algebra
Recall m x n matrix is an array of numbers with m rows and n columns
-A 1 x n matrix is a row vector.
-A n x 1 matrix is a column vector.
-The transpose of a matrix A written A^t is the n x m matri
001 - 09:50am
002 - 12:50pm
Turn off your cellphone
Write your name and student ID number in the space provided above,
and circle your instructor.
ME 270 - Spring 2015 Final NAME (Last, First):
PROBLEM 1 (20 points). Problem 1 questions are all or nothing. 50
Please show all work.
1a. Bar ABCD is loaded with a single force and couple as shown
and is held in static equilibrium by the fixed suppor
Two types of problems that can be turned into a separable or a rst order
DE problem by changing variables.
Type 1: homogeneous DE
We will be able to make this into a separable DE
Type 2: Bernoulli
We will be able to make this into a rst order DE
The idea is that U3 is redundant in list 3 - it is already a linear combination
of U1 and U2 and throwing it away does not change the span.
Fact: any linearly dependent set of vectors contains a linearly
independent subset with the same span.
A vector space is a set of objects (vectors) that sum up all that we know
Theorem: Let S be a non empty subset of a vector space V. Then S is a
subspace if and only if S is closed under addition and scalar
The inverse square matrix
Exam 1: Lessons 1-14 (recitation in two weeks)
Sometimes we are in the situation that for a given n x n matrix A, there
exists a matrix B satisfying
1. When does such a matrix B exist?
2. If B exists, how do we nd it?
Cofactors Expansion Part 2
Let A be an n x n matrix throughout
Each cofactor C Is the determinant of (n - 1) x (n - 1) matrix.
Build a matrix of cofactors, M , the (I,j) entry will be equal to C of the
original matrix A.
The ad joint matrix of A wr
Ex: Solve the system of linear equations
1. If we put augmented matrix A into REF and then use back substitution
called Gaussian elimination.
2. Put matrix A into RREF. No longer need to use substitution. This is
Properties of determinants
Fact: Let A be an n x n matrix. If A is upper triangular than the
determinant of A is - det(A) = a11a22a33ann
Elementary row operations and determinants
Let A be the original. Let B be the result of the operation:
Elementary row operations (ERO's) and row echelon matrices
Given a m x n system of linear equations. Let A be the augmented matrix
of the system.
1. Permute the equations
2. Multiply an equation by a non zero constant
3. Add a multiple of one