Consider the linear transformation T : R³ –→ R² defined by T(x1, x2, 03) = (x1 + 2x2 + 3x3, – 2x1 – 4x2 – 6æ3). - A student performs a calculation with this linear transformation and arrives at the following incorrect conclusion (blue text): N(T) = {(-2,1,0), (–3,0, 1)}, and therefore the nullspace of T is 2. Choose all of the following that are accurate statements about errors made in the student's conclusion. Please note that there may be more than one correct answer. O The solution is incorrect because (-2, 1, 0) is not in N(T). O The student's solution claims that there are only two vectors in the nullspace, but there are in fact infinitely many. O The student said that the nullspace of T is 2, but should have said that the nullity of T is 2. O The solution is incorrect because (-3,0, 1) is not in N(T). O The student said that the nullspace of T contains vectors, but this is incorrect because the nullspace should contain scalars. O The nullspace of Ţ is 1, not 2.

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Hi, I need help with this Linear Alebegra exercise, please. Thank you!

Consider the linear transformation T: R3 → R² defined by
T(*1, x2, x3) = (x1 + 2x2 + 3x3,
2x1
· 4x2
- 6æ3).
A student performs a calculation with this linear transformation and arrives at the following incorrect conclusion
(blue text):
N(T) = {(-2, 1,0), (–3,0,1)}, and therefore the nullspace of T is 2.
Choose all of the following that are accurate statements about errors made in the student's conclusion. Please
note that there may be more than one correct answer.
O The solution is incorrect because (-2, 1, 0) is not in N(T).
O The student's solution claims that there are only two vectors in the nullspace, but there are in fact infinitely many.
O The student said that the nullspace of T is 2, but should have said that the nullity of T is 2.
O The solution is incorrect because (-3, 0, 1) is not in N(T).
O The student said that the nullspace of T contains vectors, but this is incorrect because the nullspace should contain scalars.
O The nullspace of T is 1, not 2.
Transcribed Image Text:Consider the linear transformation T: R3 → R² defined by T(*1, x2, x3) = (x1 + 2x2 + 3x3, 2x1 · 4x2 - 6æ3). A student performs a calculation with this linear transformation and arrives at the following incorrect conclusion (blue text): N(T) = {(-2, 1,0), (–3,0,1)}, and therefore the nullspace of T is 2. Choose all of the following that are accurate statements about errors made in the student's conclusion. Please note that there may be more than one correct answer. O The solution is incorrect because (-2, 1, 0) is not in N(T). O The student's solution claims that there are only two vectors in the nullspace, but there are in fact infinitely many. O The student said that the nullspace of T is 2, but should have said that the nullity of T is 2. O The solution is incorrect because (-3, 0, 1) is not in N(T). O The student said that the nullspace of T contains vectors, but this is incorrect because the nullspace should contain scalars. O The nullspace of T is 1, not 2.
Expert Solution
Step 1

Given

Tx1 , x2 , x3 = x1+2x2+3x3 , -2x1-4x2-6x3

Matrix representation of this transformation is given as

123-2-4-6

Now reduce this matrix to row reduced echelon form

Consider

123-2-4-6R2R2+2R1123000

Hence we get

x1+2x2+3x3 = 0x1 = -2x2-3x3

Thus

x1x2x3 = -2x2-3x3x2x3= -210x2+-301x3

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