linear algebra midterm 1 terms

linearly dependent

at least 1 vector in the set can be written as a linear combination of the remaining vectors

linearly independent

the vectors cannot be written as a combination of the others

span

set of all possible linear combinations of vectors

homogenous matrix

Ax=0

nonhomogenous matrix

Ax=b

linearly dependent matrices dont have

pivots in every column

linearly dependent matrices have

free variables

linearly independent matrices have

pivots in every column

linearly independent matrices do not have

free variables

if there's more columns rows

the set cannot be linearly independent

consistent

has at least one solution

inconsistent

set has no solutions

why would a matrix have infinitely many solutions

system is consistent and has at least 1 free variable

a homogenous equation always has a

at least one trivial solution

trivial solution

every variable equals 0

nontrivial solution

a solution where at least one variable is not zero

homogenous set will have nontrivial solutions if

it has at least one free variable

free variable

column without a pivot

row equivalent

if one matrix can be obtained from the other using elementary row operations

ref (Row Echelon Form)

1. all zero rows are at the bottom

2. each leading entry is a nonzero to the right of the leading entry above it

3. all entires in a column below a leading entry are zeros

rref : Reduced Row Echlon Form

everything from ref +

4. each leading entry is 1

5. each leading 1 is the only nonzero number in its column

basic variables

variables corresponding to pivot columns

free variables

variables with no pivot column

if a matrix has infinitely many solutions

it is consistent

linear combination

if you can multiply vectors by a scalar and add them up to get y

identity matrix

an n x n matrix with 1's on the main diagonal and zeros every where else

transformation/function/mapping

from R^n to R^m assigns a vector to each x vector that exists in R^n

domain of T

first set of R^n

codomain of T

second set R^m

R^m

where outputs live

R^n

set of inputs

T(x) is the image of x vector under T

if x vector exists in R^n

range of T

set of all images of elements in the domain

matrix transformation

map T: R^n to R^m such that x vector becomes Ax for some m x n matrix A

standard matrix for T

unique matrix A such that T(x)=A(x) for any x that exists in R^n

onto

if every b that exists in R^m is the image of at least one x that exists in R^n

one-to-one

if each b that exists in R^m is the image of at most one x that exists in R^n

diagonal matrix

square matrix whose non-diagonal entries are all zero

zero matrix

a matrix whose entries are all zero

Aij

i = row, j = column

transpose of A

A^T --> n x m matrix whose ith column is the ith row of A