Kernel and Image

Question 1

When is a lm injective?

Answer 1

Maps distinct vectors in V to distinct vectorsin W

Question 2

When is a LM surjective?

Answer 2

If every vector in W is image of at least one vector in V

Question 3

What is the kernel of a LM and what does it mean if it contains only the zero vector?

Answer 3

Set of vectors within V that are mapped to 0w (and thus a subspace of V)
If only zero vector, map is injective meaning f does not copllapse any non-zero to zero

Question 4

What is the image of a LM and what does it mean if it contains the entire set?

Answer 4

Set of vectors in W such that w = T(v)
OR
Set of all vectors in w that can be expressed as the function f(v)

If contains entire codomain, the map is surjective, meaning f covers entire output space

Question 5

What is the nullity and rank of a LM?

Answer 5

Nullity = dim of Kernel
Rank = dim of Im

Question 6

What is the rank nullity theorem/

Answer 6

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