Define the linear transformation T by T(x) = Ax. Find ker(T), nullity(T), range(T), and rank(T). 0 -3 2 A 6. 011 (a) ker(T) (If there are an infinite number of solutions use t as your parameter.) (b) nullity(7) (c) range(T) O R2 O {(s, 0): s is any real number} O {(6s, 3t, 11s – 2t): s, t are any real number} O {(0, t): t is any real number} O R3 (d) rank(T)
Define the linear transformation T by T(x) = Ax. Find ker(T), nullity(T), range(T), and rank(T). 0 -3 2 A 6. 011 (a) ker(T) (If there are an infinite number of solutions use t as your parameter.) (b) nullity(7) (c) range(T) O R2 O {(s, 0): s is any real number} O {(6s, 3t, 11s – 2t): s, t are any real number} O {(0, t): t is any real number} O R3 (d) rank(T)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![Define the linear transformation \( T \) by \( T(x) = Ax \). Find \(\ker(T)\), \(\text{nullity}(T)\), \(\text{range}(T)\), and \(\text{rank}(T)\).
\[
A = \begin{bmatrix} 0 & -3 & 2 \\ 6 & 0 & 11 \end{bmatrix}
\]
(a) \(\ker(T)\) (If there are an infinite number of solutions use \( t \) as your parameter.)
\[
\left\{ \begin{array}{c} \end{array} \right\}
\]
(b) \(\text{nullity}(T)\)
\[
\begin{array}{c} \end{array}
\]
(c) \(\text{range}(T)\)
- \( \mathbb{R}^2 \)
- \(\{(s, 0): s \text{ is any real number}\}\)
- \(\{(6s, 3t, 11s - 2t): s, t \text{ are any real number}\}\)
- \(\{(0, t): t \text{ is any real number}\}\)
- \( \mathbb{R}^3 \)
(d) \(\text{rank}(T)\)
\[
\begin{array}{c} \end{array}
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F788d30d7-3e5d-416d-85b2-dd122fe24e5a%2F168bffa1-992e-4835-92fc-192e4708905d%2Fcrlf7l_processed.png&w=3840&q=75)
Transcribed Image Text:Define the linear transformation \( T \) by \( T(x) = Ax \). Find \(\ker(T)\), \(\text{nullity}(T)\), \(\text{range}(T)\), and \(\text{rank}(T)\).
\[
A = \begin{bmatrix} 0 & -3 & 2 \\ 6 & 0 & 11 \end{bmatrix}
\]
(a) \(\ker(T)\) (If there are an infinite number of solutions use \( t \) as your parameter.)
\[
\left\{ \begin{array}{c} \end{array} \right\}
\]
(b) \(\text{nullity}(T)\)
\[
\begin{array}{c} \end{array}
\]
(c) \(\text{range}(T)\)
- \( \mathbb{R}^2 \)
- \(\{(s, 0): s \text{ is any real number}\}\)
- \(\{(6s, 3t, 11s - 2t): s, t \text{ are any real number}\}\)
- \(\{(0, t): t \text{ is any real number}\}\)
- \( \mathbb{R}^3 \)
(d) \(\text{rank}(T)\)
\[
\begin{array}{c} \end{array}
\]
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