Linear Algebra - vector space

[jack4561]

Let V and W be two (real) vector spaces and T : V → W is a linear transformation.
Suppose Z is a subspace of W, namely Z ≤ W.
Let T −1 (Z) be the set of all vectors x in V such that T(x) lies in Z.
Namely T −1 (Z) = {x in V | T(x) lies in Z.}.


Argue that T −1 (Z) is also a subspace of V .

 

I don't know how to prove this correct

Could anyone help me

 

Thank you

 

T(V) =W > Z
......

Edited October 25, 2016 by jack4561

[Country Boy]

I would normally intepret "A- 1" for A a linear transformation as "A- I". But I suspect that you mean [math]A^{-1}[/math] (if you don't want to use Latex, write A^-1), the inverse matrix.

 

You should know that, to prove "[math]T^{-1}Z[/math] is a subspace of vector space W", you need to prove two things:

1) If u and v are in [math]T^{-1}Z[/math] then u+ v is in[math]T^{-1}Z[/math]Z.

2) if u is in [math]T^{-1}Z[/math] and a is a number, then au is in [math]T^{-1}Z[/math].

 

If u and v are in [math]T^{-1}Z[/math] then T(u) and T(v) are both in Z. [math]T(u+ v)= Tu+ Tv[/math] by definition of "linear transformation". Since Z is a subspace, Tu+ Tv is in Z. So u+ v is in [math]T^{-1}Z[/math]. You can do (2).

Edited October 25, 2016 by Country Boy