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Lecture 14

MATA23H3 Lecture Notes - Lecture 14: Scilab, Linear MapPremium


Department
Mathematics
Course Code
MATA23H3
Professor
Chrysostomou( G)
Lecture
14

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MATA23 - Lecture 14 - Properties of Linear Transformations
Properties of Linear Transformations
Example 1: Determine whether T:R2R3defined by T(x1
x2) =
2x1+ 3x2
x1
4x12x2
is a linear
transformation.
~u, ~v R2
~u =u1
u2, ~v =v1
v2
T(~u +~v) = T(u1
u2+v1
v2) = T(u1+v1
u2+v2)
=
2(u1+v1) + 3(u2+v2)
(u1+v1)
4(u1+v1)2(u2+v2)
T(~u) + T(~v) = T(u1
u2) + T(v1
v2)
=
2u1+ 3u2
u1
4u12u2
+
2v1+ 3v2
v1
4v12v2
=
2u1+ 3u2+ 2v1+ 3v2
u1+v1
4u12u2+ 4v12v2
=T(~u +~v)
T(r~u) = T(ru1
u2) = T(ru1
ru2)
=
2ru1+ 3ru2
ru1
4ru12ru2
=r
2u1+ 3u2
u1
4u12u2
=rT (~u)
Tis a linear transformation
Let Tbe a linear transformation from RnRm. Then
1. T(~
0) = ~
0
~
0is domain, ~
0codomain
2. T(α~u +β~v) = αT (~u) + βT (~v)
α, β R
1

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3. More generally, if ~v1, ~v2, . . . , ~vkare vectors in Rnand c1, c2, . . . , ckR, then
T(c1~v1+c2~v2+· · · +ck~vk) = c1T(~v1) + c2T(~v2) + · · · +ckT(~vk)
Example 2: Is T([x1, x2]) = [x1+ 1, x2]a linear transformation from R2R2
~u =u1
u2, ~v =v1
v2
T(u1
u2+v1
v2) = T(u1+v1
u2+v2) = u1+v1+ 1
u2+v2
T(u1
u2) + T(v1
v2) = u1+ 1
u2+v1+ 1
v2=u1+1+v1+ 1
u2+v2
No, it’s not
Let T:RnRmbe a linear transformation. If B=n~
b1,~
b2,...,~
bnois a basis for Rn, then
~v Rn, T (~v)is determined by T(~
b1), T (~
b2), . . . , T (~
bn)
B=n~
b1,~
b2,...,~
bnois a basis for Rn
~v Rn, ~v =r1~
b1+· · · +rn~
bn
r1R, i = 1, . . . , n
T(~v) = T(r1~
b1+· · · +rn~
bn)
Tis a linear transformation =r1T(~
b1) + · · · +rnT(~
bn)
T(~v)is determined by T(~
b1), . . . , T (~
bn)
Example 3: Let T:R2R2be a linear transformation defined by T([1,1]) = [2,3], and
T([2,1]) = [4,4]. Find T([x, y])
Let ~u =1
1, ~v =2
1
Wsubspace:
1. ~
b1,...,~
bkspan(W)
2. ~
b1,...,~
bklinearly independant
3. dim(W) = k
Any 2 are true =n~
b1,...,~
bkois a basis for W
1 2
1 11 2
01
~u, ~v are linearly independant
{~u, ~v}is a basis for R2
x
yR2,x
y=r1~u +r2~v
1 2
1 1r1
r2=x
y
2
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