Linear Midterm

Prove T: V→W is a linear map

satisfies additivity and homogeneity

satisfies additivity: T(v+w) = T(v) +T(w)

satisfies homogeneity: T(cv) = cT(v)

Define injectivity

no two x map onto same y

ker(T) ker(A)= {0}

Ax=0 has one solution (x=0)

matrix: pivot in every column (tall)

Define surjectivity

exists some x for every f(x)=y

Im(T)=F^m

Ax=b always consistent

matrix: pivot in every row (wide)

Definition of isomorphism

bijective linear map

A has an inverse (square, pivot in every row/column)

LU decomp

U = REF of A using only “add multiple of one row to another”

L = opposite row ops done in U done in reverse to the identity matrix (En^-1…E1^-1) (ex. if Ei row op is addition, Ei^-1 is the subtraction)

For Ax=b, to solve for x:

solve Ly=b

then Ux=y

Finding Eigenvalues and Eigenvectors

solve (A-λI)=0 for eigenvalues λ (non-zero elts of ker(A-λI))

solve Av=λv for eigenvectors

define eigenspace

ker(A-λI)

dimension is the number of free variables

Proving X is a subspace of Y

prove X is also a vector space (closed under addition and scalar mult) and nonempty

Proving X is a subset (⊆) of Y

prove some arbitrary element x in X is in Y

Finding the matrix to define a linear map

define the output for e1,…en where en is the number of input variables

A is the concatenation of the columns of T(e1),…T(en)

prove T: V→W is injective

let v1,v2 ∈ T, show if T(v1)=T(v2), v1=v2

Prove T: V→W is surjective

let w ∈ W, there is some v ∈ V st T(v)=w

Proving T is an isomorphism

prove T is linear

and

show T is both injective and surjective

or, show the matrix of A is invertible

Finding the inverse of a matrix A

do row ops to turn A into I

do the same row ops to I

Prove ker(T) is a subspace of the domain

1. nonempty, T(0)=0

2. closed under + (0+0=0)

3. closed under * (0×0=0)

Prove linear independence of a list (v1,…,vn)

let c1v1+…cnvn=0, show all ci must =0

Prove by counterexample that U∪W is not always a subset of V

U=[x,0], W=[0,y] is not closed under addition

Prove a system either has a unique, infinite, or no solutions

by contradiction: assume A has 2 linearly independent solutions, show all linear combinations are also solutions

Prove a system is inconsistent iff is can be row reduced to [0…0|c] c is nonzero

(→) by contradiction, assume an inconsistent system can’t be reduced to this form, there is either a pivot in every row or a free variable. Show this means the system can not be inconsistent

(←) if it has that row, show it is inconsistent (all 0 coefficients can’t result in a nonzero integer)

definition of kernel (ker(A))

all inputs for which the output is 0

prove all solutions (S)={x’+z : z is in ker(A)}

(⊆) let x ∈ S, show x ∈{x’+z : z is in ker(A)}, show x-x’ ∈ ker(A)

(⊇) let x ∈{x’+z : z is in ker(A)}, show x ∈ S, show Ax=b