The Geometry of Linear Equations¶
The fundamental problem of n linear equations in n unknowns, for example:
In this first lecture on linear algebra we view this problem in three ways.
The system above is two dimensional(
Row Picture¶
Plot the points that satisfy each equation. The interaction of the plots(if they do intersect) represents the solution to the system of equations. Looking at the figure below we see that the solution to this system of equations is
We plug this solution into the original system of equations to check our work:
The solution to a three dimensional system of equations is the common point of intersection of three planes(if there is one).
Column Picture¶
In the column picture we rewrite the system of linear equations as a single equation by turning the coefficients in the columns of the system into vectors.
Given two vector linear combination
of
Geometrically, we want to find numbers
In three dimensions, the column picture requires us to find a linear combination of three 3-dimensional vectors that equals the vector
Matrix Picture¶
We write the system of equations:
as a single equation by using matrices and vectors:
The matrix coefficient matrix
. The vector
The three dimensional matrix picture is very like the two dimensional one, except that the vectors and matrices increase in size.
Matrix Multiplication¶
How do we multiply a matrix
One method is to think of the entries of
This technique shows that
You may also calculate the product
Linear Independence¶
In the column and matrix pictures, the right hand side of the equation is a vector
for every possible vector
If the answer is "no", we say that singular matrix
. In this singular case its column vectors are linear dependent
; all linear combinations of those vectors lie on a point or line(in two dimensions) or plane(in three dimensions). The combinations don't fill the whole space.