Linear Algebra: Isomorphisms, Kernels and Images

Question 1

Isomorphisms

Answer 1

Let V and W be vector spaces.
An isomorphism from V to W is defined to be a linear transformation from V to W which is also a bijection.
If there exists an isomorphism then it is said to be isomorphic.
If V isomorphic to W, then W is also isomorphic to V

Question 2

Kernels and Images

Answer 2

Let V and W be vector spaces and T be a linear transformation from V to W.
The kernel of T is defined to be :
KerT = {u E V | T(u) = 0}

The image of T is defined to as:
ImT = {T(u) | u E V}

hence KerT is the preimage of {0} under T and ImT is the range of T.

Question 3

General Properties of Kernels and Images

Answer 3

t can be shown that kernels & images satisfy certain general properties.
* For a linear transformation T from a vector space V to a vector space W ,
four such properties are:
(1) Ker T is a subspace of V .

(2) Im T is a subspace of W .

(3) T is injective if & only if KerT = {0} (where0 is the zero of V ).

(4) T is surjective if & only if Im T = W .