Download M392 Lecture April 6th

M392 Lecture April 6th
A linear expression in variables x,y,z is a sum of constants times those variables.
Example: 3x + 2y + 7z is a linear expression.
A linear equation sets a linear expression equal to a constant.
Example: 3x + 2y + 7z is a linear equation.
A system of linear equations is a list of equations that are to be solved
simultaneously for the unknowns (variables)
Example: Here’s a linear system of 3 equations in 3 variables
3x + 7y + 8z = 41
2x + y
=4
4x
+z=7
The simplest useful linear systems consist of 2 equations in 2 variables.
For example, in school you used to solve equations by rewriting:
2x + 3y = 5 -----> times 3 ------>6x+9y = 15
3x + 4y = 7 -----> times 2 ------>6x +8y = 14
__________
Subtract to get
y=1
Plug back into first eq:----->2x +3(1) = 5 ----> x = 1
Solution x = 1; y = 1
We say that the system has ONE solution.
However, other systems behave differently
The system below has NO SOLUTIONS. The equations contradict each other!
2x + 3y = 5 -------> times 2------>4x + 6y = 10 …….10 doesn’t equal 7!!
4x + 6y = 7------------------------->4x + 6y = 7
The system below has MANY SOLUTIONS. The equations repeat each other.
2x + 3y = 5 -------> --------------->2x + 3y = 5
4x + 6y = 10-----divide by 2----->2x + 3y = 5
In other words, this system of 2 equations in 2 unknowns is the same as the
following very simple system: 1 equation, 1 unknown:
2x + 3y = 5
There are many ways to write down the (infinitely) many solutions. Here’s the one
we use.
Let y = s be any number. Then x = (5-3s)/2
For example, when s = 0, y = 0 and x = 5/2
When s =1, y = 1 and x = 1 ,,, and so on.
Each of the three kinds of systems of 2 equations in 2 unknowns can be solved
graphically, by sketching the graphs of the equations, which are straight lines in the
x,y plane. Solutions are pairs (x,y) where the two lines meet.
Case I. Two lines cross at a point: ONE SOLUTION
Case II: Two lines are parallel: NO SOLUTIONS
Case III: Two lines are identical: INFINITELY MANY SOLUTIONS
Heuvers Chapter I: Matrices (plural of matrix) are used to simplify the bookkeeping when we solve systems of linear equations.
An m x n matrix is a rectangular array of numbers with m rows, n columns.
 3 4 5
 and B =
Example: A = 
 1 2 3
 4 4 4

 are 2 x 3 matrices.
 3 3 3
1 2


C =  3 4  is a 3 x 2 matrix.
 2 2


MATRIX ALGEBRA
Multiply a matrix by a scalar ( a number) : Multiply every matrix entry by that
number.
 3 4 5   6 8 10 
 = 

2A = 2 
1
2
3
2
4
6

 

Add two matrices: Only possible when they have the same size
For example, A and B are both size 2 x 3, and so they can be added
 3 4 5
 4 4 4
7 8 9
 + 
 = 

A + B = 
 1 2 3
 3 3 3
 4 5 6
Exercise: Find 2A + 3B
Multiplying and dividing matrices are tricky operations. Be patient…..
Matrix entries: we write
For example
aij for the entry in row i, column j of matrix A.
a11  3; a23  3; c21  3;
The transpose of matrix M is written M T , obtained by writing the rows of M as
columns of a new matrix M T .
In words: Row i , column j of M T is row j, column i of M
T
In symbols:
ij
ji .
M
M
7 4


7 8 9
T
 , then M =  8 5 
Example: if M = 
 4 5 6
9 6


A row matrix is a matrix with one row.
A column matrix is a matrix with one column.
The transpose of a column matrix is a row matrix, and vice versa.
If
1
 
C   3. 
4
 
R
T
then C T 
1
3 4
Recall that the dot product of ( 1 3 5) and (2 4
1 3 5  2 4 6  1 2  3  4  5  6  2  12  30
6) is
Finally, we can define matrix multiplication.
When you multiply A by B, the product is written AB
For matrices, the product is defined only when
The number of columns of A equals the number of rows of B
In particular the product AA is defined only when A is square: same number of
rows and columns.
Note: you can’t multiply a 2 x 3 matrix by a 2 x 3 matrix.
However, you can multiply a 2 x 3 matrix by a 3 x 2 matrix
 3 4 5
 B =
Example: A = 
 1 2 3
1 2


 3 4  Find the product M = AB
 2 2


Multiplication Rule for matrices: If M = AB, then
The entry in row i, column j of M is the dot product of
Row i of A with the transpose of Column j of B
For example row 1, column 1 of the product AB is
( 3 4 5)  (1 3 2) = 3  1 + 4  3 + 5  2 = 3 + 12 + 10 = 25