Introduction to Matlab

1703 Lecture: Introduction to Matlab

Matrix Representation

$ x - 2y = 1 $

Can someone give me some values of x and y that satisfy this equation?

x	y
3	1
1	0
2	$ \frac{1}{2} $
101	50

How many combinations of x and y will satisfy this equation? $ \infty $

How else can we represent this equation?

As a line:

(insert figure here)

Is the slope positive or negative?

What's the slope?

$ 3x + 2y = 11 $

Some values of x and y that satisfy this equation?

x	y
3	1
1	4
75	-107

How many combinations of x and y will satisfy this equation? $ \infty $

We can also represent this equation as a line

(insert figure here - both lines on same plot)

What's the slope?

We can see that the point x=3, y=1 is where these two lines meet - which means it is the combination of x and y that satisfies both of these equations.

How else can we represent these equations?

In column form:

$ \left[ \begin{array}{cc} 1 \\ 3 \end{array} \right] x + \left[ \begin{array}{cc} -2 \\ 2 \end{array} \right] y = \left[ \begin{array}{cc} 1 \\ 11 \end{array} \right] $

After pushing these two columns together, we get:

$ \left[ \begin{array}{cc} 1 & -2 \\ 3 & 2 \end{array} \right] \left[ \begin{array}{cc} x \\ y \end{array} \right] = \left[ \begin{array}{cc} 1 \\ 11 \end{array} \right] $

What about the equations:

$ \begin{eqnarray} 3x + 5y + z & = & 16 \\ 2x - y - z & = & 3 \\ x + 4y - 2z &=& 3 \end{eqnarray} $

How to represent this graphically?

Instead of lines, use planes

Let's look at the first 2 equations only:

(Insert figure here)

The two planes intersect to form a line

Now the third equation: Also a plane

The line that represents the intersection of these first two equations will intersect the third equation's plane at one point

We can also represent these equations in matrix form:

$ \left[ \begin{array}{cc} 3 \\ 2 \\ 1 \end{array} \right] x + \left[ \begin{array}{cc} 5 \\ -1 \\ 4 \end{array} \right] y + \left[ \begin{array}{cc} 1 \\ -1 \\ -2 \end{array} \right] z = \left[ \begin{array}{cc} 16 \\ 3 \\ 3 \end{array} \right] $

$ \left[ \begin{array}{cc} 3 & 5 & 1 \\ 2 & -1 & -1 \\ 1 & 4 & -2 \end{array} \right] \left[ \begin{array}{cc} x \\ y \\ z \end{array} \right] = \left[ \begin{array}{cc} 16 \\ 3 \\ 3 \end{array} \right] $

How do we represent 4 equations graphically?

5 equations?

6 equations?

We run into a limit using graphical methods

What about the matrix representation of 4 equations? 5 equations? 6 equations?

The matrix representation is really easy