21 Dugh-л/4√2).]at maps eithat leaves=/4 radianhorizontalizontal x₁-XI.that trans-nd then re-Cal x₂-axiscts pointsgh the X2-r transfor-= the angleely a rota-ion?that T (e₁)Using the15.16.????????I??X1?HX2?X3X1X2==3x14x1x₁ - x2 + x3đi – 12-2x1 + x₂XIIn Exercises 17-20, show that T is a linear transformation byfinding a matrix that implements the mapping. Note that x₁, x₂,...are not vectors but are entries in vectors.17. T(X1, X2, X3, X4) = (0, x₁ + x2, x2 + x3, x3 + x^)18. /T(x1, x₂) = (2x2 – 3x₁, x₁ − 4x2, 0, x₂)-19. T(X1, X2, X3) = (x₁ - 5x₂ + 4x3, x2 - 6x3)20. T(X1, X2, X3, X4) = 2x₁ + 3x3 - 4x4 (T: R → R)21. Let T R2 R2 be a linear transformation such thatT(X1, X₂) = (x₁ + x2, 4x₁ + 5x₂). Find x such that T(x) =(3,8).=→>>>22. Let T: R2 R³ be a linear transformation such thatT(x₁, x₂) = (x₁ - 2x2, -x1 + 3x2, 3x1 - 2x2). Find x suchthat T(x)= (-1,4,9).In Exercises 23 and 24, mark each statement True or False. Justifyeach answer.23. a. A linear transformation T: R" → R" is completely de-termined by its effect on the columns of the n x n identitymatrix.->>b. If T: R² R2 rotates vectors about the origin throughan angle , then T is a linear transformation.c. When two linear transformations are performed one afteranother, the combined effect may not always be a lineartransformation.d. A mapping T: R" → R" is onto R" if every vector x inR" maps onto some vector in Rm.e. If A is a 3 x 2 matrix, then the transformation x→ Axcannot be one-to-one.

To solve for x such that T(x) = (3,8), we need to find values of x1 and x2 such that T(x1, x2) =…

Engineering

Computer Science

21

Database System Concepts

7th Edition

ISBN:9780078022159

Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan

Chapter1: Introduction

Section: Chapter Questions

Problem 1PE

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Question

Dugh-л/4
√2).]
at maps ei
that leaves
=/4 radian
horizontal
izontal x₁-
XI.
that trans-
nd then re-
Cal x₂-axis
cts points
gh the X2-
r transfor-
= the angle
ely a rota-
ion?
that T (e₁)
Using the
15.
16.
?
?
?
?
?
?
?
?
I
?
?
X1
?
H
X2
?
X3
X1
X2
=
=
3x1
4x1
x₁ - x2 + x3
đi – 12
-2x1 + x₂
XI
In Exercises 17-20, show that T is a linear transformation by
finding a matrix that implements the mapping. Note that x₁, x₂,...
are not vectors but are entries in vectors.
17. T(X1, X2, X3, X4) = (0, x₁ + x2, x2 + x3, x3 + x^)
18. /T(x1, x₂) = (2x2 – 3x₁, x₁ − 4x2, 0, x₂)
-
19. T(X1, X2, X3) = (x₁ - 5x₂ + 4x3, x2 - 6x3)
20. T(X1, X2, X3, X4) = 2x₁ + 3x3 - 4x4 (T: R → R)
21. Let T R2 R2 be a linear transformation such that
T(X1, X₂) = (x₁ + x2, 4x₁ + 5x₂). Find x such that T(x) =
(3,8).
=
→>>>
22. Let T: R2 R³ be a linear transformation such that
T(x₁, x₂) = (x₁ - 2x2, -x1 + 3x2, 3x1 - 2x2). Find x such
that T(x)= (-1,4,9).
In Exercises 23 and 24, mark each statement True or False. Justify
each answer.
23. a. A linear transformation T: R" → R" is completely de-
termined by its effect on the columns of the n x n identity
matrix.
->>
b. If T: R² R2 rotates vectors about the origin through
an angle , then T is a linear transformation.
c. When two linear transformations are performed one after
another, the combined effect may not always be a linear
transformation.
d. A mapping T: R" → R" is onto R" if every vector x in
R" maps onto some vector in Rm.
e. If A is a 3 x 2 matrix, then the transformation x→ Ax
cannot be one-to-one.

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