- #1
stunner5000pt
- 1,461
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If X and Y are nonempty subsets of vector space V such taht span X = span Y = V , must there be a vector common to both X and Y? Justify.
my intuition thinks there need not be one, but i can be wrong!
to go about proving it is where the fun starts!
[tex] a_{1} x_{1} + a_{2} x_{2} + ... + a_{n} x_{n} = b_{1} y_{1} + ... + b_{n} y_{n} = V [/tex]
suppose X contaiend only one vector then ax = by. That seems to say taht one is a scalar multiple of the other
suppose X contaiend two vectors then
[tex] a_{1} x_{1} + a_{2} x_{2} = b_{1} y_{1} + b_{2} y_{2} = V [/tex]
here that means the resultant of the first addition yields the resultant of the second addition. However it doesn't mean that X = Y since ai is not bi for all i.
So there need not be any common vector
is this right? Can you help with formalizing it?
Thank you!
my intuition thinks there need not be one, but i can be wrong!
to go about proving it is where the fun starts!
[tex] a_{1} x_{1} + a_{2} x_{2} + ... + a_{n} x_{n} = b_{1} y_{1} + ... + b_{n} y_{n} = V [/tex]
suppose X contaiend only one vector then ax = by. That seems to say taht one is a scalar multiple of the other
suppose X contaiend two vectors then
[tex] a_{1} x_{1} + a_{2} x_{2} = b_{1} y_{1} + b_{2} y_{2} = V [/tex]
here that means the resultant of the first addition yields the resultant of the second addition. However it doesn't mean that X = Y since ai is not bi for all i.
So there need not be any common vector
is this right? Can you help with formalizing it?
Thank you!