1553 Linear Algebra GT

Linear Independence

if Ax=0 has ONLY the trivial solution

Linear Transformations

Ax=b

Domain

R^n (input)

Codomain

R^m (output)

Range

set of all possible b's for Ax=b

span of transformation matrix = {T.A}

One-to-ONe

For every b in R^m, there is only one x

1 input -> 1 output

Unique solution

Pivot in every COLUMN

Onto

For every b in R^m there exist at least one x

Range = codomain

Pivot in every ROW

Ax=b

Given A and x, how do you find b?

[A|B] = x

Given A and b, how do you find x?

Subspace H

-includes zero vector

-includes sum of vectors (u+v)

-includes scalars of vectors (cu)

ColA

set of all possible linear combinations of A

= span{A}

NulA

set of all solutions to Ax=0

Dimension

# of vectors in basis

Dim(ColA)

rank

Dim(NulA)

# of free variables

Rank

# of pivot columns

[A|1]

inverting an nxn matrix

x=(A^-1)b

Given A and b, find x

Diagonalizable

if A is an nxn matrix that has n linearly independent eigenvectors

Geometric Multiplicity

dim(eigenspace)

Linear Combination

sum of scaled vectors

c1v1+c2v2+...+cnvn

Transpose

an nxm matrix turned into an mxn matrix by turning the columns into rows and vice versa

Inverse

Ax=b has one solution, AB=1 and BA=1

Determinant

-row replacements do not change outcome

-scaled by same scale of the matrix

-row swaps scale outcome by -1

Eigenvalue

a number in R such that Av=(this #)v has a nontrivial solution

Eigenspace

set of all eigenvectors of A with the same eigenvalues, plus the zero vector

Orthogonal

perpendicular

x dot y = 0

Orthonormal

each pair of vectors is orthogonal and each vector is a unit vector

x bar parallel is

the orthogonal projection of x bar onto the line

two vectors are orthogonal to each other if

u * v = 0

magnitude is

the square root of each component squared

In an LU decomposition, the columns of L are

the pivotal columns in original matrix

In an LU decomposition, U is

the reduced echelon form of A

length is

the magnitude squared

a unit vector is

a vector whose length is 1

to diagonalize a 3x3 matrix,

find P and D. P is the matrix of eigenvectors, D is the coreesponding matrix with eigenvalues in diagonals

b is a linear combination of a if

there are solutions to the augmented matrix

consistent means

there are one or more solutions to the system

inconsistent means

there are no solutions to the system

a solution set with one free variable is a ____, two free variables is a ______

line, plane

the additive property of linear transformations

T(x+y)=T(x)+T(y)

the scalar property of linear transformations

T(cx)=cT(x)

linearly independent has _____ solutions

trivial

a linearly dependent system has ______ solutions

nontrivial

to find the determinant of a 3x3 matrix using diagonals

downward products - upward

to find the determinant of a 3x3 matrix using reducing

multiply the diagonals of the echelon form matrix

lambda is an eigenvalue if A-(lambda)I is a linearly _______ matrix

dependent

basis for ColA is given by

the pivotal columns

dimension for ColA

number of pivotal columns

basis for NulA is given by

soln vectors to Ax=0

dimension of NulA

number of vectors in the basis for NulA

equation for RankA

RankA+dim of NulA=n

algebraic multiplicity

the number of times it is an eigenvalue

geometric multiplicity

the number of eigenvectors for an eigenvalue

to find the Inverse of a 3x3 matrix,

set up [A I] and reduce to [I A]

how to find if a vector (u) is in the Nul space of A

if Au=0, then it is in the Nul space

how to find a vector in ColA

just a column of A

how to find a vector in NulA

a soln vector in terms of the free variable

cosine of angle between vectors v and u

(uv)/(IuIIvI)

scalar component of u in direction of v

(u*v)/IvI

implicit equation

y = mx + b, a line in a plane

parametric form

coordinates where one point is in terms of the other variables

(x,y) = (t, 1-t)

linear equation

a1x1+a2x2... = b

system of linear equations

collection of linear equations

consistent linear system

a system of equations with solutions

equivalent linear systems

2+ linear systems with the SAME solution set

m x n matrix

m rows and n columns

augmented matrix

matrix with solutions on one side

elementary row operations

scale, swap, row replacement

row equivalent

one matrix can be obtained from the other using row operations, have the same solution set

inconsistent matrix

the last column (the augmented column) is a pivot column - no solution to the linear system

Rⁿ

the set of ordered lists of n numbers

row echelon form

all zero rows at the bottom

first non zero in each row is farther right as you go down

below a leading entry its all zeroes

pivot

first nonzero number of a row

pivot column

column that contains a pivot IF matrix is in row echelon form

reduced row echelon form

row echelon + all pivots are 1 + pivot columns only contain the pivot

free variable

if the corresponding column is not a pivot column

parametric form

write matrix solution as system of linear equations, and move all free variables to the right of =

possible number of solutions to a matrix

zero, one, infinite

vector

coordinates in Rⁿ drawn as an arrow - length and direction, not location

properties of vectors

all regular addition and multiplication properties

vector addition

head to tail

vector subtraction

tail to tail

linear combination of vectors

combining vectors into one equation to produce a new vector

span

all the linear combinations of a set of vectors - "creates" the plane or the space

the set of solutions to Ax = 0

matrix equation

Ax = b

Ax = b has a solution if and only if...

b is in the span of the columns of A = b is a linear combination of v1, v2, vn

a solution always exists IF...

Ax = b has a solution for all b

the span of A is ALL of Rm

A has a pivot in every row

homogenous

set of linear equations where Ax=0

inhomogenous

Ax = b

trivial solution

x vector = 0

non trivial solution

non zero solutions - IF there is a free variable = a column with no pivot

parametric vector form

list all equations from reduced matrix including free varaiables, and move free variables to the right of = to make a vector equation

solution set of Ax = 0

creates the span

solution set is a line

one free variable

solution set is a plane

two free variables

homogenous vs nonhomogeous solutions

vector associated with free varable stays the same but an additional translation vector is added

= translation of a span

column span

span of the columns of A

all b that makes Ax = b consistent

solution set

all x so that Ax = b

b is fixed (different than column span)

linear independence

only has the trivial solution (only constant that relates the vectors is 0)

each vector is independent of the others

matrix A has a pivot in every column!!