Linear Independence and Transformations

Question 1

What is the definition of linear independence ?

Answer 1

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

Question 2

If the zero vector is apart of {v1,..,vk}, what can we say about its dependence ?

Answer 2

Vectors {v1,…,vk} are linearly independent

Question 3

What can we say about the dependence of {v1,…,vk} if one of them is a linear combination of another ?

Answer 3

Vectors {v1,…,vk} are linearly dependent

Question 4

What can we say about the dependence of vectors {v1,…,vk} in a matrix A if the REF OF A has non-zero rows?

Answer 4

The non-zero rows of A are linearly independent

Question 5

What is the definition of a basis?

Answer 5

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

Question 6

If {v1,…,vk} are a basis of V, what can we say about every other vector v in V?

Answer 6

Every vector v in V can be written as a linear combination of the basis vectors {v1,…,vk}

Question 7

What can we say about the non-zero rows of a echelon matrix A in regards to basis and row spaces?

Answer 7

The non-zero rows in the echelon form are a basis of the row space

Question 8

When is the row space of a matrix the entire field F to power n?

Answer 8

when the REF equals the identity matrix

Question 9

What is the definition of a finite dimension of V?

Answer 9

V is finite dimensional if V is spanned by a finite set

Question 10

If {v1,…,vk} span V and V is finite dimensional, what can we say about {v1,…,vk}
What if they are linearly independent?

Answer 10

• 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

Question 11

What is the definition of dimension ?

Answer 11

The common size of all the bases of a vector space is called the dimension, denoted dim(V)

Question 12

What are the three things vectors {v1,…,vk} must be at least two of to be a basis?

Answer 12

• Linearly independent
• A spanning set of V
• k = dim(V)

Question 13

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

Answer 13

True

Question 14

What can we say about row rank and column rank and the ranks with respect to row operations?

Answer 14

• Row Rank = Column Rank

- Row operations do not change the column rank of a matrix

Question 15

For a matrix A, what is the definition of Rank(A)

Answer 15

Rank(A) = column rank - Row rank

Question 16

What are the definitions of :
Row rank
Column rank

Answer 16

Row rank - the dimension of the row space

Column rank - the dimension of the column space

Question 17

What is the definition of a linear mapping?

Answer 17

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

Question 18

How would you find the matrix of T with respect to a standard bases of the transformation?

Answer 18

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

Question 19

True or False:

Linear Transformations are uniquely determined by its action on a basis

Answer 19

True

Question 20

What is the definition of the image of a linear transformation ?

Answer 20

Im(T) = {T(v):vεV}

The solution space of the transpose of the Mat(T)

Question 21

What is the definition of the rank of a linear transformation?

Answer 21

Rank = dim(Im(T))

Question 22

What is the definition of the Kernal of a linear transformation ?

Answer 22

Ker(T) = {vεV:T(v)=0}

The solution space of the homogeneous system of Mat(T)

Question 23

What is the definition of the nullity of a linear transformation?

Answer 23

Null(T)=dim(Ker(T))

Number of parameters in the general solution to AX=0

Question 24

How do you find a basis of the image of Mat(T)?

Answer 24

Transpose Mat(T) and row reduce to REF
The rows will be the results basis of the Image

Question 25

How do you find the basis of the Kernal of Mat(T)?

Answer 25

Solve the solution space of the homogeneous equation of Mat(T) = A
Set AX=0

Question 26

What is the rank nullity theorem ?

Answer 26

For a Linear Transformation T:V->W

dim(V)=rank(T)+null(T)