Definitions and Theorems

Sunday, February 05, 2012

4:59 PM



Vector addition

Scalar multiplication


Linear combination


Theorem 1.1.1




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.



Standard Basis






Theorem 1.2.1

(Prove non-empty set, vector addition, and scalar multiplication)


Theorem 1.2.2



Theorem 1.3.1


Dot product


Theorem 1.3.2






Unit Vector


Theorem 1.3.3 (Vector Properties)





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


System of linear equations

(Note: A linear equation can be represented geometrically as a hyperplane.)





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.



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:

  1. Multiply a row by a non-zero scalar.
  2. Add a multiple of a row to another row.
  1. Swap two rows

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:

  1. All rows that have at least non-zero entry are above rows with only zeros.
  1. The leftmost non-zero entry in a row is a 1, called a leading one.
  1. The leading one in a row is to the right of the leading one in any row above it.
  1. Any column containing a leading one has it as its only non-zero entry.



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



The rank of a matrix is the number of leading ones in the RREF of the matrix.


Theorem 2.2.4



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