Show that the xy plane W=(x,y,0) in R3 is generated by (i) u=[120] and v=[010]...

Linda Seales

Linda Seales

Answered

2022-01-05

Show that the xy plane W=(x,y,0) in R3 is generated by
(i) u=[120] and v=[010] (ii) u=[210] and v=[130]

Answer & Explanation

Louis Page

Louis Page

Expert

2022-01-06Added 34 answers

Let (x,y,0) be any element from W. For a and b in R, we have
(x,y,0)=a(1,2,0)+b(0,1,0)
=(a,2a+b,0)
Comparing both side we get
x=a and y=2a+b
Now from y=2a+b
b=y2a
=y2x
Therfore, for any (x,y,0) in W, we can write
(x,y,0)=x(1,2,0)+(y2x)(0,1,0)
(x,y,0){(1,2,0),(0,1,0)}
Therefore, W is generated by u and v.
Let (x,y,0) be any element from W. For a and b in R, we have
(x,y,0)=a(2,1,0)+b(1,3,0)
=(2a+b,a+3b,0)
Comparing both side we get
x=2a+b ...(1)
and
y=a+3b ...(2)
Now we solve equationn (1) and (2) to find the value of a and b. From (1) and (2), we get
x+2y=2a+b2a+6b=7b
b=x+2y7
Now from (2)
a=y+3b
=y+3(x+2y7)
=y+3x+6y7
=7y+3x+6y7
=3x+13y7
a=3x+13y7
Therfore, for any (x,y,0) in W, we can write
(x,y,0)=a(2,1,0)+b(1,3,0)
=3x+13y7(2,1,0)+x+2y7(1,3,0)
(x,y,0){(2,1,0),(1,3,0)}
Therefore, W is generated by u and v.

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