Extension By Linearity

Question 1

What is a linear map uniquely determined by

Answer 1

Kinear map is uniquely determined by it action on a basis

Question 2

What is extension of linearity prop

Answer 2

Extension of linearity prop is:
If V,W are vector spaces of F, V1,…,VN a basis of V and W1,…Wn are any vectors in W, exists a unique phi in L(V,W) s.t phi(Vi) = Wi
Any vector can be written by taking a function of basis vectors

Question 3

When is phi from linearity prop a bijection

Answer 3

Phi from linearity prop is a bijection when W1,…,Wn is a basis of W

Question 4

What is rank nullity theorem

Answer 4

Rank-nullity theorem is:
If phi :V to W is linear and V is finite dim then dim( im(phi)) + dim (ker(phi)) = Dim V
Rank phi + nullity of phi

Question 5

What is phi if ker(phi) = 0

Answer 5

If ker(phi) = 0, phi is injective, and surjects E if V finite Dim

Question 6

If phi: V to W is I near and dim V = dim W, what is phi and when is this false

Answer 6

If phi: V to W is linear and dim V = dim W, phi is an isomorphism
This is false for infinite dim V,W