Linear Algebra!!

Pivot

The first non-zero entry in a non-zero row of a matrix that has been brought to row echelon form.

Row echelon form

All zero rows are at the bottom & Each leading pivot is to the right of the pivot above it & every thing below each pivot is 0

Reduced row echelon form

Each pivot must be 1 & and the only non zero number in it’s column

Pivot variables

The variables corresponding to pivot columns

Non pivot variables

There is no pivot in the entire column of this variable

Row Equivalence

Two matrices are row equivalent if you can get from one to the other using elementary row operations.

Gauss-Jordan elimination

You use row operations to turn the matrix into RREF.

Pivot Parametization

is how you write the solution when you have free variables.

Rouché capelli theorem (PROOF)

mainly used to determine whether a system has a solution and how many solutions it has.

1. To determine solutions from systems of equations

1. Set up augmented matrix

2. Reduce to row echelon form

3. Then uses rouche capelli theory to determine solutions

If there is a pivot in every column in the coefficient matrix

Unique solution

There is a free(non pivot) variable in the coefficient matrix

Infinite solution

There is a pivot in the augmented column

No solution

The last row is all zeroes in the augmented matrix

Infinite solution

VECTORS

an ordered list of numbers that represents a quantity with magnitude and direction.

Linear combination of vectors

when you multiply vectors by scalars and add them together.

2v - 3v2, where 2 & 3 are scalars

The span of vectors, v1, v2,…..,Vn in R^m

Is the set of all linear combinations of v1,….Vn

A set of vectors is linearly dependent if..

at least one vector can be written as a linear combination of the others.

• no trivial solution

• a column w/o a pivot present in coefficient matrix

A set of vectors is linearly independent if

the only way to get the zero vector is by using all zero coefficients

• pivot in every column

• Trivial solution

The dot product of vectors u & v is the scalar…

U*V=U1V1+…..UnVn

Scalar

The NORM of the vector u is defined by

||u|| = √(v * u)

• its length/magnitude.

The distance between vectors U and V is defined by

D(U , V) = || U - V || & NORM OF DISTANCE

Orthogonal- we can say that vectors u & v are orthogonal if

Vector U * vector V = 0

Vector equations and systems

The vector equation (X1V1+….XnVn = W) has the same solution set as the augmented matrix

(V V1 V2 | w)

Linear Dependence Theorem (PROOF)

If one vector in a set is a linear combination of the other vectors, then the set is linearly dependent otherwise, independent.

One vector is a multiple of another

Linearly dependent

4. One vector is a combination of the others

Dependent

There are 4 vectors but only 3 dimensions. N > M in R^m

Dependent

There is a 0 in the vector

Dependent