Let V and W be vector spaces with subspaces V 1 andW 1 , respectively. If T: V--
ID: 2940345 • Letter: L
Question
Let V and W be vector spaces with subspaces V1 andW1, respectively. If T: V--->W is linear,prove that T(V1) is a subspace of W and that{xV: T(x)W1} is a subspace of V. I understand the first part of this problem, but I reallyreally really need the second part by tonight!!! The notationwith the brackets is very confusing to me, and I'm not sure how toapproach the problem. I will rate anyone a LIFESAVER if they answer tonight!!! :-)! I understand the first part of this problem, but I reallyreally really need the second part by tonight!!! The notationwith the brackets is very confusing to me, and I'm not sure how toapproach the problem. I will rate anyone a LIFESAVER if they answer tonight!!! :-)!Explanation / Answer
Let U={v in V| T(v) is in W1}. We wish to show that U isa subspace of V. It suffices to show the following: (i) 0 is in U. (ii) If u,v are in U, then so is u+v. (iii) For any scalar if u is in U, then so is u. (i) is trivial. Suppose u,v are in U. Then by definition, T(u),T(v) are inW1. Since W1 is a subspace of W, it followsthat T(u) + T(v) is also in W1. Since T is linear,T(u)+T(v)=T(u+v). Hence T(u+v) is in W1 and so bydefinition of U, u+v is also in U. This proves (ii). Finally, for any constant , again as before, by linearity ofT, T(u)=T(u). Since u is in U, T(u) is inW1. Since W1 is a subspace, and T(u) is inW1 so is T(u) => T(u) is in W1and that proves again that u is in U. This proves what youwant.
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