# S,T: V onto W are linear maps

by adottree
Tags: linear, maps
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 P: 4 S,T: V onto W are both linear maps. Show that M:={x out of V s.t. Sx out of Range(T)} is a subspace of V I know that to show M is a subspace of V I must show: i. 0 out of M ii. For every u, v out of M, u+v out M iii. For every u out of M, a out of F, au out of M. I just don't know how to start it, can someone help?
 Math Emeritus Sci Advisor Thanks PF Gold P: 39,538 You are using "out of" where I would use "in" but I understand what you need. First, as I have pointed out before. You do not need to show that 0 is in M. Since 0v= 0 (the 0 vector), show that M is closed under scalar multiplication immediately gives you that. M is (in my language!) the set of all vectors, x, in V such that Sx is also in the range of T: there exist some y in V such that Sx= Ty. Okay, suppose x[sub]1[/sup] and x2 are in M- that is, there is y2 in V such that Sx1= Ty1 and y2 such that Sx2= Ty2. What can you say about S(x1+ x2). Now suppose x is in M- that is, there is y in V such that Sx= Ty- and a is in F. What can you say about S(ax)?
 P: 4 S(x1 + x2) = S(x1) + S(x2) (this is because S is a linear map) = T(y1) + T(y2) = M(x1) + M(x2) and S(ax) = aS(x) = aT(y) = aM(x) Thanks for your help, is this kind of right?
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,538
S,T: V onto W are linear maps

 Quote by adottree S(x1 + x2) = S(x1) + S(x2) (this is because S is a linear map) = T(y1) + T(y2) = M(x1) + M(x2) and S(ax) = aS(x) = aT(y) = aM(x) Thanks for your help, is this kind of right?
You should not be saying "M(x1)", "M(x2)", or "M(x)" since they are meaningless. M is not a linear map, it is a subspace of V.
 P: 4 S(x1 + x2) = S(x1) + S(x2) (this is because S is a linear map) = T(y1) + T(y2) therefore (x1 +x2) in M and S(ax) = aS(x) = aT(y) therefore (ax) in M I think this is better (I hope)! Thanks you've been a great help!!!
 P: 230 just to make it complete you should write: S(x1 + x2) = S(x1) + S(x2) (this is because S is a linear map) = T(y1) + T(y2) = T(y1+y2) so you can se that S(x1+x2) is in the range of T, namly hit by y1+y2 under T.

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