jack4561 Posted October 25, 2016 Share Posted October 25, 2016 (edited) 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 Link to comment Share on other sites More sharing options...
Country Boy Posted October 25, 2016 Share Posted October 25, 2016 (edited) 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 Link to comment Share on other sites More sharing options...
Recommended Posts
Create an account or sign in to comment
You need to be a member in order to leave a comment
Create an account
Sign up for a new account in our community. It's easy!
Register a new accountSign in
Already have an account? Sign in here.
Sign In Now