Exercises: Find the Matrix Representing a Given Linear Map

Problem 1

For each of the linear maps specified below find the matrix representing it.

  1. \(L\from \RNr\to \RNrSpc{2}\) transforms the unit vector in \(\RNr\) into the vector \((–5,7)\) in \(\RNrSpc{2}\).

  2. \(L\from \RNrSpc{2}\to \RNrSpc{2}\) transforms the unit vectors \(\StdBssVec{1}\) and \(\StdBssVec{2}\) into the vectors \((3,0)\) and \((0,3)\), respectively.

  3. \(L\from \RNrSpc{3}\to \RNrSpc{2}\) transforms the unit vectors \(\StdBssVec{1}\), \(\StdBssVec{2}\), and \(\StdBssVec{3}\) of \(\RNrSpc{3}\) into the vectors \((3,1)\), \((4,2)\), and \((-2,-3)\), respectively

  4. \(L\from \RNrSpc{1}\to \RNrSpc{4}\) transforms \(-2\) into the point \((2,-1,1,4)\).

Problem 2

Consider the function \(f\from \RNrSpc{2}\to \RNrSpc{2}\), \(f(x,y)=(x^2-y^2,2xy)\).

  1. Compute the \((2,2)\)-matrix

    \[ A \DefEq \left[ \begin{array}{cc} \uparrow & \uparrow \\ f(1,0) & f(0,1) \\ \downarrow & \downarrow \end{array} \right] \]

  2. Compute the vectors

    \[ f(1,1)\qquad \text{and}\qquad A \left[ \begin{array}{c} 1 \\ 1 \end{array} \right] \EqDef \Vect{a} \]

  3. Explain how it is possible that \(f(1,1)\neq \Vect{a}\).