(1) Define f : R---> R by f(x) = mx+b a. Show that f is a linear tranformation w
ID: 2966685 • Letter: #
Question
(1) Define f: R---> R by f(x) = mx+b
a. Show that f is a linear tranformation when b = 0
b. Find the property of a linear when b DOES NOT equal 0
(Hint: Look up and use the definition of a linear fransformation)
(2) Show that the transformation T: R2 ---> R3 defines by T
equal to
is linear by showing it satifies the two properties given by in the defintion of linear transformation.
(3) Let T: R3 ---> R3be a linear transformation. Let {v1, v2, v3} be a set of linearly vectors in R3. Show that the set {T(v1), T(v2), T(v3)} is also linearly dependent.
(Hint: Use the definition of linearly dependent to write down an equation using the v's and then apply T to both sides and use the properties of linear transformations)
x1 x2Explanation / Answer
1.
(a)
f is a linear transformation
=>
f(kx) = kf(x) for all k
=>
mkx +b = mkx +bk for all k
=>
b = bk for all k
=>
b(k-1) = 0 for all k
=>
b= 0
thus proved
(b)
when bis not equal to zero
both the properties f(kx)= kf(x), f(x1+x2) = f(x1)+f(x2) are not satisfied
2.
T(x1,x2) = (x1+x2, 3x2, x1-x2)
T(kx1,kx2) = (kx1+kx2, 3kx2, kx1-kx2) = k(x1+x2,3x2,x1-x2) = kT(x1,x2)
T(x1,x2)+f(y1,y2) = (x1+x2,3x2,x1-x2)+(y1+y2, 3y2,y1-y2) = ((x1+y1)+(x2+y2), 3(x2+y2), (x1+y1)-(x2+y2)) = T(x1+y1,x2+y2)
the above two properties prove that T is a linear transformation
3.
let
aT(v1)+bT(v2)+cT(v3) = 0
=>
T(av1+bv2+cv3) = 0= T(0)
=>
av1+bv2+cv3 = 0, but we know that v1,v2,v3 are linearly independent
=>
a= b = c = 0
=>
T(v1), T(v2), T(v3) are linearlt independent
thus proved
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.