Linear Independence and Transformations Flashcards
(26 cards)
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
