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Linear Algebra Review 1 (Solutions (Geometric view (Vector Equations (We…
Linear Algebra Review 1
General
3 ways to work with systems of linear equations
Vector Equations
\(x_1a_1+x_2a_2+x_na_n=b\)
The augmented matrix is
[\(\quad a_1\quad a_2\quad ... \quad a_n\quad | \quad b\quad \)]
x=weights(constants that stretch or shrink the vector)
a
=vector
b
=vector (Linear combination of the left side vectors)
Example
Matrix Equations
A
x
=
b
Augmented matrix is also
[\(\quad a_1\quad a_2\quad ... \quad a_n\quad | \quad b\quad \)]
a=columns of A matrix
b
=vector
A=m x n matrix (rows x columns)
x
=vector containing real numbers
b
=vector as a result of matrix multiplication
Ordinary system of linear equations
\(a_1x_1+a_2x_2+a_3x_3+a_nx_n=b\)
a=any real or complex number
x=variable(Can't be multiplied by another variable,squared,etc)
b=any real number
Definitions
Solutions
Geometric view
System of Linear Equations
We view the augmented systems as planes or lines
Each row represents a line or plane
If a system has a solution it means at some point the planes or lines cross
If Unique crosses path only once
There is one and only point that satisfies the equation
Infinite means the planes are completely overlapping
If Infinite every point satisfies each equation
1 more item...
If no solution the planes never cross
The equations are not satisfied at any point
Vector Equations
We view the augmented system as vectors
Each column represents a vector
What we are really solving is
\(x_1a_1+x_2a_2+x_na_n=b\)
If there is a solution this means
b
is a linear combination of the left side
If there are no solutions this means
b
vector is not in the span of {\(a_1,a_2...a_n\)}
The left side represents linear combinations
Linear combinations represent all the possibilities of adding vectors and all their possible scalar multiples
Span
The linear combinations result in a line or plane
Imagine each result of adding a vector with a random scalar multiple as a point
All these possible results, results in countless combinations which result in a plane or line
Matrix Equations
We view matrix A as an acting on the
x
and it makes a new vector
b
. We call it a transformation
As we can see the transformation in the picture transforms
u
into a different vector with different dimensions
It can be viewed as a function. In a typical function input is equal to some output.
f(x) is the output when x is the input
T(x) is the output vector when input vector is in the input
3 Possibilites
Consistent-Unique
There is an answer
This is the only possible answer
Occur when the last column in an augmented matrix isn't a pivot column
Consistent-Infinitely many solutions
There is a solution
But an infinite number of solutions exists
Occurs when we have at least one free variable
Inconsistent-No solution
Has a row that takes the form
[0 0 0 ... 0 | b]
Where b is nonzero
Makes no sense to have 0=b
where b is nonzero
0=5 shows an impossible scenario
