Proving (x_k+1, . . . , x_n) forms a basis for V/kerT

[HyperbolicMan]

Hi, I was working through this proof in my linear al textbook and there's this one step I can't get past. Any help would be appreciated.

Homework Statement

Let V be a finite dimensional vector space, and let T be a linear map defined on V.

ker T $$\subseteq$$ V and I am T $$\cong$$ V/kerT

Let (y_1, . . . , y_k) be a basis of ker T. Augment this list by (x_k+1, . . . , x_n) to a basis for V: (y_1, . . . , y_k, x_k+1, . . . , x_n).

Now here's the part that's getting me:

"Obviously, (x_k+1, . . . , x_n) forms a basis of V/kerT"

Homework Equations

ker T $$\subseteq$$ V and I am T $$\cong$$ V/kerT

The Attempt at a Solution

I believe that span(x_k+1, . . . , x_n) is isomorphic to V/kerT (they have the same dimension), but I don't see how (x_k+1, . . . , x_n) actually forms a basis for V/kerT

 

[hgfalling]

Well x_n is linearly independent and it has the same dimension as V/ker T. So it's a basis.

 

[HyperbolicMan]

Well x_n is linearly independent and it has the same dimension as V/ker T. So it's a basis.

yes, but I think this only shows that (x_k+1, . . . , x_n) is basis for span(x_k+1, . . . , x_n), not for V/kerT.

 

[Outlined]

Follows straight from the definition of basis:

1. $$x_{k+1} + \ker{T}, \dots , x_{n} + \ker{T}$$ are linearly independent in $$V / \ker{T}$$;
2. $$x_{k+1} + \ker{T}, \dots , x_{n} + \ker{T}$$ span whole $$V / \ker{T}$$.

Tell me which part you have problems with.