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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

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter6: Linear Transformations
Section6.CR: Review Exercises
Problem 21CR: Let T be a linear transformation from R2 into R2 such that T(4,2)=(2,2) and T(3,3)=(3,3). Find...
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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.
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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