3) Let T : R? → R³ be the linear transformation such that T(< 1,2 >) =< 1, 1,1 > and T(< 2,3 >) =< 0,2, 5 > a) Find the standard matrix for T b) Find a basis for the orthogonal complement of the range of T
3) Let T : R? → R³ be the linear transformation such that T(< 1,2 >) =< 1, 1,1 > and T(< 2,3 >) =< 0,2, 5 > a) Find the standard matrix for T b) Find a basis for the orthogonal complement of the range of T
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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The question is in the picture please answer a and b
![### Linear Transformation Problem
**Problem Statement:**
3) Let \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^3 \) be the linear transformation such that
\[ T(<1,2>) = <1,1,1> \]
and
\[ T(<2,3>) = <0,2,5> \]
**Tasks:**
a) Find the standard matrix for \( T \).
b) Find a basis for the orthogonal complement of the range of \( T \).
**Solutions:**
For part (a), we need to determine the standard matrix \( [T] \) of the linear transformation \( T \). The matrix can be found by determining how \( T \) maps the standard basis vectors of \( \mathbb{R}^2 \).
For part (b), to find a basis for the orthogonal complement of the range of \( T \), we first need to find the range space of \( T \) and then determine its orthogonal complement within \( \mathbb{R}^3 \).
**Steps for part (a)**:
1. Express the given transformation conditions in matrix form.
2. Solve the system of equations to find the matrix representation of \( T \).
**Steps for part (b)**:
1. Determine the range of \( T \).
2. Find the orthogonal complement of the range within \( \mathbb{R}^3 \).
Feel free to explore linear algebra topics around transformations, basis, and orthogonal complements in the relevant concept sections for a deeper understanding and practice problems.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcecab317-fab7-42bc-8c65-e82787a88e59%2F889cdfc3-6847-48ef-ba55-2f24e4a09bb8%2Fm5ja5hh_processed.png&w=3840&q=75)
Transcribed Image Text:### Linear Transformation Problem
**Problem Statement:**
3) Let \( T : \mathbb{R}^2 \rightarrow \mathbb{R}^3 \) be the linear transformation such that
\[ T(<1,2>) = <1,1,1> \]
and
\[ T(<2,3>) = <0,2,5> \]
**Tasks:**
a) Find the standard matrix for \( T \).
b) Find a basis for the orthogonal complement of the range of \( T \).
**Solutions:**
For part (a), we need to determine the standard matrix \( [T] \) of the linear transformation \( T \). The matrix can be found by determining how \( T \) maps the standard basis vectors of \( \mathbb{R}^2 \).
For part (b), to find a basis for the orthogonal complement of the range of \( T \), we first need to find the range space of \( T \) and then determine its orthogonal complement within \( \mathbb{R}^3 \).
**Steps for part (a)**:
1. Express the given transformation conditions in matrix form.
2. Solve the system of equations to find the matrix representation of \( T \).
**Steps for part (b)**:
1. Determine the range of \( T \).
2. Find the orthogonal complement of the range within \( \mathbb{R}^3 \).
Feel free to explore linear algebra topics around transformations, basis, and orthogonal complements in the relevant concept sections for a deeper understanding and practice problems.
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