Linear Independence and Transformations Flashcards
What is the definition of linear independence ?
Vectors v1,…,vk are Linearly independent if the only solution to the equation
α1v1+α2v2+…+αkvk=0 is
α1=α2=…=αk=0
Otherwise, Vectors v1,…,vk are linearly dependent
If the zero vector is apart of {v1,..,vk}, what can we say about its dependence ?
Vectors {v1,…,vk} are linearly independent
What can we say about the dependence of {v1,…,vk} if one of them is a linear combination of another ?
Vectors {v1,…,vk} are linearly dependent
What can we say about the dependence of vectors {v1,…,vk} in a matrix A if the REF OF A has non-zero rows?
The non-zero rows of A are linearly independent
What is the definition of a basis?
A basis is a linearly independent spanning set
eg: unit vectors i,j,k -> (1,0,0),(0,1,0),(0,0,1) are a standard basis
If {v1,…,vk} are a basis of V, what can we say about every other vector v in V?
Every vector v in V can be written as a linear combination of the basis vectors {v1,…,vk}
What can we say about the non-zero rows of a echelon matrix A in regards to basis and row spaces?
The non-zero rows in the echelon form are a basis of the row space
When is the row space of a matrix the entire field F to power n?
when the REF equals the identity matrix
What is the definition of a finite dimension of V?
V is finite dimensional if V is spanned by a finite set
If {v1,…,vk} span V and V is finite dimensional, what can we say about {v1,…,vk}
What if they are linearly independent?
- If {v1,…,vk} span V a subset of {v1,…,vk} will be a basis of V
- If they are linearly independent, {v1,…,vk} can be extended to a basis of V
What is the definition of dimension ?
The common size of all the bases of a vector space is called the dimension, denoted dim(V)
What are the three things vectors {v1,…,vk} must be at least two of to be a basis?
- Linearly independent
- A spanning set of V
- k = dim(V)
True or False:
n vectors {v1,…,vn} in matrix A are a basis of F^n if and only if det(A) does not equal 0
True
What can we say about row rank and column rank and the ranks with respect to row operations?
- Row Rank = Column Rank
- Row operations do not change the column rank of a matrix
For a matrix A, what is the definition of Rank(A)
Rank(A) = column rank - Row rank
What are the definitions of :
Row rank
Column rank
Row rank - the dimension of the row space
Column rank - the dimension of the column space
What is the definition of a linear mapping?
A linear mapping is defined as
T:V->W if
T is closed under addition and scalar multiplication
A linear mapping takes all the vectors in the space V and transforms them into the vector space W
How would you find the matrix of T with respect to a standard bases of the transformation?
Apply T to the first standard basis and then express the answer as a linear combination of the second standard basis
Repeat this for all the bases
Put the coefficients into a matrix as the columns
True or False:
Linear Transformations are uniquely determined by its action on a basis
True
What is the definition of the image of a linear transformation ?
Im(T) = {T(v):vεV}
The solution space of the transpose of the Mat(T)
What is the definition of the rank of a linear transformation?
Rank = dim(Im(T))
What is the definition of the Kernal of a linear transformation ?
Ker(T) = {vεV:T(v)=0}
The solution space of the homogeneous system of Mat(T)
What is the definition of the nullity of a linear transformation?
Null(T)=dim(Ker(T))
Number of parameters in the general solution to AX=0
How do you find a basis of the image of Mat(T)?
Transpose Mat(T) and row reduce to REF The rows will be the results basis of the Image
How do you find the basis of the Kernal of Mat(T)?
Solve the solution space of the homogeneous equation of Mat(T) = A
Set AX=0
What is the rank nullity theorem ?
For a Linear Transformation T:V->W
dim(V)=rank(T)+null(T)
