Rank and Nullity

Question 1

dimension of V

Answer 1

If B is a basis of V then |B| is the dimension of V

Question 2

kernel of f:V->W

Answer 2

the subspace {v∈V: f(v)=0}

Question 3

Image of f: V->W

Answer 3

the subspace {f(v)∈W: v∈V}

Question 4

nullity

Answer 4

the dimension of the kernel of f

Question 5

rank

Answer 5

the dimension of the image of f

Question 6

proposition 6.2. surjective/injective linear maps. f:V->W

Answer 6

f is surjective iff imf = W

f is injective iff kerf = {0}

Question 7

proposition 6.3. Rank nullity Formula.

Answer 7

r(f)+n(f)=dimV

Question 8

quick proof to show f is linear

Answer 8

1. write f(v) as a matrix multiplication

2. since left matrix multiplication is a linear map then f(v) is a linear map.

Question 9

?How to calculate rank?

Answer 9

1. we know r(f)≤dim(im(f))

2. starting at the biggest possible rank show im(f) has a result of this dimension