Given sets U, W of pts in R^3, find finite set for W cutting U

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yossa

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Hi everyone,
I need linear help.


Given (R ^ 3):

. . . . .\(\displaystyle U\, =\, \mbox{Span}\left\{(1,\, 1,\, 2),\, (2,\, 2,\, 1)\right\}\)

. . . . .\(\displaystyle W\, =\, \mbox{Sp}\left\{(1,\, 3,\, 4),\, (2,\, 5,\, 1)\right\}\)

Find a finite finite set for W cutting U


Thanks :)
 

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U is the space of all vectors of the form a(1, 1, 2)+ b(2, 2, 1)= (a+ 2b, a+ 2b, 2a+ b) and V is the space of all vectors of the form p(1, 3, 4)+ q(2, 5, 1)= (p+2q, 3p+ 5q, 4p+ q). I presume that by "W cutting U" you mean the intersection of the two spaces. If v is in both spaces then v= (a+ 2b, a+ 2b, 2a+ b)= (p+ 2q, 3p+ 5q, 4p+ q) for some a, b, p, and q.

We must have a+ 2b= p+ 2q, a+ 2b= 3p+ 5q, and 2a+ b= 4p+ q, three equations to solve for a, b, p, and q. Since there are four unknowns but only three equations we can solve for three of the unknowns in terms of the other one- the intersection is one dimensional. Subtracting the first equation from the second, 0= 2p+ 3q so q= (-2/3)p. So (p+ 2q, 3p+ 4q, 4p+ q)= (p- (4/3)p, 3p- (8/3)p, 4p- (2/3)p)= ((-1/3)p, (1/3)p, (10/3)p)= p/3(-1, 1, 10) . That one dimensional space is the span of the single vector (-1, 1, 10).
 
HallsofIvy said:
U is the space of all vectors of the form a(1, 1, 2)+ b(2, 2, 1)= (a+ 2b, a+ 2b, 2a+ b) and V is the space of all vectors of the form p(1, 3, 4)+ q(2, 5, 1)= (p+2q, 3p+ 5q, 4p+ q). I presume that by "W cutting U" you mean the intersection of the two spaces. If v is in both spaces then v= (a+ 2b, a+ 2b, 2a+ b)= (p+ 2q, 3p+ 5q, 4p+ q) for some a, b, p, and q.

We must have a+ 2b= p+ 2q, a+ 2b= 3p+ 5q, and 2a+ b= 4p+ q, three equations to solve for a, b, p, and q. Since there are four unknowns but only three equations we can solve for three of the unknowns in terms of the other one- the intersection is one dimensional. Subtracting the first equation from the second, 0= 2p+ 3q so q= (-2/3)p. So (p+ 2q, 3p+ 4q, 4p+ q)= (p- (4/3)p, 3p- (8/3)p, 4p- (2/3)p)= ((-1/3)p, (1/3)p, (10/3)p)= p/3(-1, 1, 10) . That one dimensional space is the span of the single vector (-1, 1, 10).
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Thanks my friend, you really helped me :)
 
hey ppl, give love, life is sweet and beautiful but in the end come to an end, and sometimes really fast .....
 
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