I am reviewing an old exams and need help figuring out the problem\'s I previous
ID: 2901747 • Letter: I
Question
I am reviewing an old exams and need help figuring out the problem's I previously help.
1. Let T; R3 --> R3 be a transformation defined by T(x1,x2, x3) = (x1, -x2, x3).
(a) Prove that T is a linear transformation.
(b) Find the standard matrix of T.
(c) Is T onto R3? Explain.
(d) Is T one-to-one? Explain.
2. Let T: Rn --> Rm be a linear transformation. Assume that {T(v1), T(v2), T(v3)} are linearly dependent vectors in Rm, but {v1, v2, v3} are linearly independent vectors in Rn. Prove that T is not "1-1".
I would appreciate an explanation for this as I'm trying to understand the material and learn this myself. Thank you!
Explanation / Answer
1.
let (x1,x2,x3), (y1,y2,y3) belongs to R^3
=>
T(x1,x2,x3) +T(y1,y2,y3) = (x1,-x2,x3)+(y1,-y2,y3) = (x1+y1, -x2-y2, x3+y3) = T(x1+y1, x2+y2, x3+y3)
kT(x1,x2,x3) = k(x1,-x2,x3) = (kx1,-kx2,kx3) = T(kx1,kx2,kx3)
=>
above two statements prove that T is a linear transformation
(b)
matrix =
[1 0 0
0 -1 0
0 0 1]
(c)
T(x1,-x2,x3) = (x1,x2,x3)
=>
T is onto
(d)
let T(x1,x2,x3) = T(y1,y2,y3)
=>
(x1,-x2,x3) = (y1,-y2,y3)
=>
x1 = y1, x2= y2, x3 = y3
=>
T is one-one
2.
T(v1),T(v2),T(v3) are linearly dependent
=>
aT(v1)+bT(v2) +cT(v3) = 0 for some scalars a,b,c
=>
T(av1+bv2+cv3) = 0 = T(0)
av1+bv2+cv3 !=0 since v1,v2,v3 are linearly independent
and T(av1+bv2+cv3) = T(0)
=>
T is not one-one
thus proved
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