- #1
Hurricane3
- 16
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Homework Statement
let T: R[tex]^{3}[/tex] -> R[tex]^{3}[/tex] be the mapping that projects each vector x = (x(subscript 1) , x(subscript 2) , x(subscript 3) ) onto the plane x(subscript 2) = 0. Show that T is a linear transformation.
Homework Equations
if c is a scalar...
T(cu) = cT(u)
T(u + v) = T(u) + T(v)
The Attempt at a Solution
Well I don't know if I proved the first condition correctly, but I have:
T(cx) = T(c (x (subscript 1) , x (subscript 2) , x (subscript 3) )
= T (c x (subscript 1) , c x (subscript 2) , c x (subscript 3) )
= ( c x (subscript 1) , c 0 , c x (subscript 3) )
= c T (x)
Did I do this correctly??
And same with the second condition... I have:
T ( x + 0 ) = T ( (x (subscript 1), x (subscript 2) , x (subscript 3) ) + ( 0 , 0 , 0 ) )
= T ( x(subscript 1) , x (subscript 2) , x (subscript 3) ) + T ( 0 , 0 , 0)
= ( x(subscript 1) , 0 , x (subscript 3) ) + ( 0 , 0 , 0)
= T(x) + T(0)
Did I do this correctly as well??
And my conclusion is that this is a linear transformation...
Oh and sorry about the x(subscript #)... i tried using the Latex Reference thing, but it shows the subscripts as superscripts...
Thanks