Definitions and Theorems
Sunday, February 05, 2012
4:59 PM
Points 
Vector addition 

Scalar multiplication 
Linear combination 
Theorem 1.1.1
Span Spanning set 

Vector Equation 
Theorem 1.1.2
Linearly Dependent Linearly Independent Trivial Solution 
Theorem 1.1.3
In order for a spanning set to be as small as possible, it must be linearly independent.
Basis 

Standard Basis 
Kplane 

Theorem 1.2.1
(Prove nonempty set, vector addition, and scalar multiplication)
Theorem 1.2.2
Theorem 1.3.1
Dot product 
Theorem 1.3.2
Length Norm 
Unit Vector 
Theorem 1.3.3 (Vector Properties)
Orthogonal 

Orthogonal set 
A set of vectors is an orthogonal set if every pair of vectors in the set is orthogonal. 
Cross Product 
Theorem 1.3.4 (Properties of Cross Product)
Linear equation 

Coefficients 

System of linear equations 
(Note: A linear equation can be represented geometrically as a hyperplane.) 
Solution 

Consistent Inconsistent 
A system of linear equations is consistent if there is at least one solution. Otherwise it is inconsistent. 
Theorem 2.1.1
If the following system of linear equations:
(A consistent system with more than one solution has infinitely many solutions.)
Solution set 
The set of all solutions of a system of linear equations is the solution set of the system. 
Equivalent 
If 2 systems of linear equations have the same solution set, then they are equivalent. 
Augmented Matrix 
For the system of linear equations:
Is the coefficient matrix.
Is the augmented matrix.
Note: Rows represent equations and columns represent variables. 
Elementary row operations (EROs) 
There are 3 elementary row operations (EROs) for solving a system of linear equations:

Row reducing matrix 
Applying EROs to a matrix is called row reducing a matrix. 
Row equivalent 
If there is a sequence of EROs that transform one matrix to another, then the matrices are row equivalent. 
Theorem 2.2.1
If augmented matrix are row equivalent, then the corresponding system of linear equations are equivalent.
Reduced row echelon form (RREF) 
A matrix is in reduced row echelon form (RREF) if:

Theorem 2.2.2
Every matrix has a unique RREF.
Free variable 
Homogeneous system 
A system of linear equations is homogeneous if the RHS contains only zeros. 
Theorem 2.2.3
Solution space 
The solution set of a homogeneous system is called the solution space of the system 
Rank 
The rank of a matrix is the number of leading ones in the RREF of the matrix. 
Theorem 2.2.4
Created by Tim Pei with Microsoft OneNote 2010
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