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Section 3.4 Isomorphisms

Definition 3.4.1.

A function \(F:\mathbb{L}_1\longrightarrow \mathbb{L}_2\) is an Isomorphism if \(F\) is a linear transformation that is \(1-1\) and maps onto \(\mathbb{L}_2\text{.}\)

Given that \(T: \mathbb{R}^3 \longrightarrow \mathbb{R}^2\) is a linear transformation such that \(T(1,1,0) = (1,2)\text{,}\) \(T(0,0,1)=(0,-1)\) and \(T(0,1,0) =(2,0)\text{.}\) Determine \(T(3,3,0)\text{,}\) \(T(1,2,3)\text{,}\) and \(T(x,y,z)\text{.}\)

Describe the problems that could occur if the set on which the linear transformation is defined is:

  1. not a spanning set for \(\mathbb{R}^3\)

  2. not linearly independent

Use a system of equations to:

  1. write (5,4,3) as a linear combination of (1,1,1),(0,1,1) and (0,0,1)

  2. write \(5+4x +3x^2\) as a linear combination of \(1+x+x^2\text{,}\) \(x+x^2\) and \(x^2\)

Define a linear transformation from \(\mathbb{P}_3\) to \(\mathbb{R}^3\text{.}\) Is it 1-1? Is it onto? What is its inverse transformation?