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S,T: V onto W are linear maps 
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#1
Mar808, 08:00 PM

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? 


#2
Mar808, 09:20 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,339

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 x_{2} are in M that is, there is y_{2} in V such that Sx_{1}= Ty_{1} and y_{2} such that Sx_{2}= Ty_{2}. What can you say about S(x_{1}+ x_{2}). 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)? 


#3
Mar808, 09:51 PM

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? 


#4
Mar808, 10:17 PM

Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 39,339

S,T: V onto W are linear maps



#5
Mar808, 10:43 PM

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!!! 


#6
Mar908, 07:29 AM

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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