In Exercises 25-28, determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer.25. The transformation in Exercise 17 24esn1In Exercises 15 and 16, fill in the missing entries of the matrix,assuming that the equation holds for all values of the variables.15.16.? ?? ??22???????XID-LX3X1X2=3x₁ - 2x34x1x1 - x2 + x3X1 X2-2x1 + x₂X1In Exercises 17-20, show that T is a linear transformation byfinding a matrix that implements the mapping. Note that x₁, x..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) = (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: R4 → 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 o, then T is a linear transformation,

Given: Tx1, x2, x3, x4=0, x1+x2, x2+x3, x3+x4. To Check:Whether the given transformation is one-one…

Answered: 17. T(X₁, X2, X3, X4) = (0, X₁ + X2, X2…

Math

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17. T(X₁, X2, X3, X4) = (0, X₁ + X2, X2 + X3, X3 + X4)

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter9: Systems Of Equations And Inequalities

Section9.7: The Inverse Of A Matrix

Problem 32E

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Question

In Exercises 25-28, determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer. 25. The transformation in Exercise 17

2
4
es
n
1
In Exercises 15 and 16, fill in the missing entries of the matrix,
assuming that the equation holds for all values of the variables.
15.
16.
? ?
? ?
?
22
?
?
?
?
?
?
?
XI
D-L
X3
X1
X2
=
3x₁ - 2x3
4x1
x1 - x2 + x3
X1 X2
-2x1 + x₂
X1
In Exercises 17-20, show that T is a linear transformation by
finding a matrix that implements the mapping. Note that x₁, x..
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) = (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: R4 → 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 o, then T is a linear transformation,

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