[image]

where [image] are coefficients

[image]

where [image] is a constant

[image]

[image]

.

.

.

[image]

• Row Swapping - one row may be swapped with another
• Scaling - both sides of an equation may be multiplied by a non-zero constant
• Row Combination - an equation may be replaced by the sum of itself and a multiple of another equation

• no eqaution may be multiplied by zero
• can't add a multiple of a row to itself
• can't swap a row with itself

A system is in echelon form if each leading variable is to the right of the one above it and all-zero rows are at the bottom.

Example:

[image]

When any step in Gaussian Elimination produces a contradictory equation.

For example,

[image]

0 [image] 2 is a contradictory equation

When we reach echelon form, with each variable a leading variable in its row.

For example,

[image]

When we reach echelon form without any contradictory equations and at least one variable is not a leading variable in its row.

For example,

 [image]

In an echelon form linear system, non-leading variables are free.

For example, in a system with three unknowns, x, y, z

[image]

a, in a non-leading position, indicates z is a free variable.

The leading variables are described in terms of the free variables.

For example, the solution set of a linear system with 4 unknowns (x,y,z,w) and 2 free variables (z,w) would describe x and y in terms of z and w as in this solution set:

[image]

The variables need to describe a family of solutions.

For example, only two variables (z,w) may be needed to describe a linear system with 4 unknowns; so the solution would have two parameter variables, z and w.

A matrix is an m x n rectangular array of numbers with m rows and n columns. 

Each number in a matrix is called an entry.

An uppercase letter is used to denote a matrix.

The lowercase of the same letter, with row and column indices, is used to denote an entry in the same matrix.

For example, a matrix is denoted by A while an entry in the matrix is denoted by [image]

A vector (or column vector) is a matrix with a single column. A matrix with a single row is a row vector. (may also be called linear arrays)

The entries in a vector are called its components (may also be called coordinates, entries or elements).

What notation is used to define a vector?

A lowercase letter overlined by an arrow.

For example,

[image]

where u and v are both vectors.

To add two vectors, both must have the same number of components.

[image]

Basic properties of vector addition:

1. [image]
2. [image]
3. [image]
4. [image]

Multiply each vector component by the scalar.

[image]

Basic properties of vector scalar multiplication (where [image]) :

1. [image]
2. [image]
3. [image]
4. [image]

If k < 0, vector is scaled in the opposite direction.

[image]

where [image] is any particular solution and

where the number of vectors [image] equal the number of free variables in the system after Gaussian Elimination

A linear system is homogeneous when it has a constant of zero and can be written as:

[image]

Yes. They always have at least one, the zero-vector. Some may have many solutions.

The general solution to a linear system includes both the particular ( [image] ) and the homogeneous ( [image] ) solutions of the system.

A matrix with the same number of rows and columns.

A square matrix whose entries are the coefficients of a homogeneous linear system with a unique solution (the system reduces to echelon form with no free variables or contradictory equations).

A square matrix with many solutions.

Note: We normally expect a square matrix to have a unique solution; when it doesn't, it is singular (i.e. unusual, not what we expected)

The length of a vector [image] is the square root of the sum of the squares of its components.

[image]

For any non-zero vector [image], the vector [image]/[image] has a length of one. We say that [image]normalizes [image].

The dot product is the scalar result of the linear combination by multiplication of two n-component real vectors.

[image]

Basic properties of dot product:

1. [image]
2. [image]
3. [image]
4. [image]

Note: the dot product of a vector with itself is the length of the vector squared: [image].

If the dot product of two vectors is 0 then the vectors are orthogonal (perpendicular to each other).

For any [image], [image] with equality if and only if one of the vectors is a non-negative multiple of the other.

For any [image], [image] with equality if and only if one vector is a scalar multiple of the other.

Gauss elimination transforms a matrix into echelon form, Gauss-Jordan reduces echelon form to row canonical form.

Gauss elimination places zeros in the columns below the leading variables (pivots), Gauss-Jordan places zeros in the columns above the leading variables.

Gauss-Jordan reductionfollows the same rules as Gauss elimination but begins from the bottom, right corner of the matrix. It is only applied to columns above pivots (i.e. columns holding free variables are not zeored out.)

A matrix or linear system is in reduced echelon form if, in addition to being in echelon form, each leading entry is a one and it is the only non-zero entry in it's column.

Example of a matrix in reduced row echelon form:

[image]

• Any matrice reduces to itself
• If A reduces to B then B reduces to A (symmetry)
• If A reduces to B and B reduces to C then A reduces to C (transivity)

Two matrices that are interreducible by row operations are row equivalent.

Notation: A ~ B

Apply Gauss-Jordan to both matrices, if they produce the same reduced echelon form then they are row equivalent and can be derived from each other.

Note: matrices of different sizes cannot be row equivalent.

A inear combination  of linear combinations is always a linear combination.

Lemma:

In an echelon form matrix, no non-zero row is a linear combination of the other non-zero rows.

A vector u is a unit vector if [image] for any non-zero vector [image]

For vectors [image] and [image] [image] the Euclidean distance is

[image]

To determine if a square matrix is invertible, append it's identity matrix and reduce to echelon form. If the left hand side of the resulting matrix is in upper triangular form, the matrix is invertible. To find the inverse, continue row echelon form; the left-hand side will now be the identity matrix and the right-hand side will be the inverse of the original.

Example:

[ 1 1 2] [ 3 0 3] [-2 3 0] [1 0 0] [0 1 0] [0 0 1]

[ 1 1 2| 1 0 0] [ 3 0 3| 0 1 0] [-2 3 0| 0 0 1]

becomes, after reduction,

[ 1 0 0| -3 2 1] [ 0 1 0| -2 4/3 1] [ 0 0 1| 3 -5/3 -1]