21 leavesradianrizontalntal x₁-at trans-then re-1 x₂-axists pointsh the X2-transfor-the anglethat T (e₁)Using the?16. ??1???ely a rota-ion? 201XI[*]-X2x₁-x₂-2x1 + 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 + X4)18. / T (X1, X2)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: R¹ → 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 that22. X→T(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 p, 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 R™e. If A is a 3 x 2 matrix, then the transformation x → Axcannot be one-to-one.24. a. Not every linear transformation from R" to R" is a matrixtransformation.standard matrix for a linear transfor-

Answered: X2, 3, 4 2X1 + 3X3 4X4 21. Let T: R2 R2…

X2, 3, 4 2X1 + 3X3 4X4 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).

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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Concept explainers

Use of XOR function

XOR function or exclusive OR function was introduced in Excel 2013. This function is available under excel logical functions. For the given statement, the XOR function will return true only when either of the statements is true. It returns false when both the statements are true or when both the statements are false.

Question

leaves
radian
rizontal
ntal x₁-
at trans-
then re-
1 x₂-axis
ts points
h the X2-
transfor-
the angle
that T (e₁)
Using the
?
16. ?
?
1
?
?
?
ely a rota-
ion? 201
XI
[*]-
X2
x₁-x₂
-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 + X4)
18. / T (X1, X2)
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: R¹ → 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
22. X
→
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 p, 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 R™
e. If A is a 3 x 2 matrix, then the transformation x → Ax
cannot be one-to-one.
24. a. Not every linear transformation from R" to R" is a matrix
transformation.
standard matrix for a linear transfor-

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