- #1
HyperbolicMan
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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.
Let V be a finite dimensional vector space, and let T be a linear map defined on V.
ker T [tex]\subseteq[/tex] V and I am T [tex]\cong[/tex] 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"
ker T [tex]\subseteq[/tex] V and I am T [tex]\cong[/tex] V/kerT
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
Homework Statement
Let V be a finite dimensional vector space, and let T be a linear map defined on V.
ker T [tex]\subseteq[/tex] V and I am T [tex]\cong[/tex] 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 [tex]\subseteq[/tex] V and I am T [tex]\cong[/tex] 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