Linear Algebra
Solving Linear Equations
Vectors and linear equations
Aim of linear algebra?
To find a solution of a system of linear equations
What is an equation?
Exactly, an equation with unknowns (one or more unknowns)
What is linear?
No product
Linear Algebra
Vector spaces and subspaces
The nullspace of : solving
Example
Find the null space of .
Solution
free variables
To obtain independent special solutions, use independent cases for free variabels, s
uch as and
and
Rank
More and more equa
Linear Algebra
Vector spaces and subspaces
Spaces of vectors
Definition
Let be a nonempty set. is called a field if
and for all
The following properties hold:
such that for all
such that
such that
such that
,
Spaces of vectors
Definition
Let be a
Linear Algebra
Solving Linear Equations
Elimination using matrices
Process to solve a system
Back-substitution!
Elimination using matrices
Row echelon form
A matrix is a row echelon form if
1) Any row consisting of zeros are at the bottom.
2) In each n
Linear Algebra
Orthogonality
Orthogonal bases and Gram-Schmidt
Definition
The vectors are orthonormal if
Note
Let be a matrix whose column vectors are orthonormal.
since the -entry of is .
If is square, then . is called an orthogonal matrix.
even w
Linear Algebra
Orthogonality
Least square approximations
When does a system have no solution?
Let .
If , then has no solution.
In this case, it may be true that # of columns of # of rows of .
Find to minimize
Such is called a least square solution.
In
Linear Algebra
Solving Linear Equations
Inverse matrices
Definition
A matrix is invertible if there is a matrix such that
The matrix is called the inverse matrix of , denoted by .
For example,
If is not invertible, is called to be singular.
Inverse m