Exercises: The Effect of Linear Transformations on Objects

Problem 1

For the linear transformation \(L\from \RNrSpc{1}\to \RNrSpc{2}\), \(L(x)=(-2x,3x)\), describe its effect on the given geometrical objects in the domain.

  1. What does \(L\) do to the \(0\)-vector of \(\RNrSpc{1}\)?

  2. How does \(L\) transform the standard unit vector \(\StdBssVec{1}\) of \(\RNrSpc{1}\)?

  3. How does \(L\) transform the vector \(\Vect{x}=2\)?

  4. What is the image under \(L\) of the vector \(\Vect{x}=10\)?

  5. What is \(L(-4)\)?

  6. What is the image under \(L\) of the domain \(\RNrSpc{1}\)?

Problem 2

About the linear transformation \(L\from \RNrSpc{1}\to \RNrSpc{3}\), \(L(t)=(t,-t,2t)\) answer the following questions.

  1. What is the domain of \(L\)?

  2. What does \(L\) do to the \(0\)-vector of \(\RNrSpc{1}\)?

  3. How does \(L\) transform the standard unit vector \(\StdBssVec{1}\) of \(\RNrSpc{1}\)?

  4. What is \(L(-5)\)?

  5. What is the image under \(L\) of the unit interval \([0,1]\) of \(\RNrSpc{1}\)?

Problem 3

Determine the image of the inclusion functions

\[\CoordInclsn{1},\CoordInclsn{2}\from \RNr \longrightarrow \RNrSpc{2}\]
Problem 4

Determine the effect of the coordinate projections

\[\PrjctnOnto{1},\PrjctnOnto{2}\from \RNrSpc{2}\longrightarrow \RNr\]

on the specified objects of \(\RNrSpc{2}\).

  1. The point \((7,-1)\)

  2. The rectangle with corners \((2,2)\), \((5,2)\), \((2,10)\), \((5,10)\).