Exercises: A Linear Map from a Matrix

Problem 1

For each of the statements below indicate if it is true or false.

  1. The matrix \(\Mtrx{A}\) representing a linear function \(L\from \RNrSpc{7}\to \RNrSpc{5}\) has size \((5,7)\).

  2. The matrix \(\Mtrx{A}\) representing a linear function \(L\from \RNrSpc{7}\to \RNrSpc{5}\) has size \((7,5)\).

  3. The matrix \(\Mtrx{A}\) representing a linear function \(L\from \RNrSpc{7}\to \RNrSpc{5}\) is square shaped with \(7\) rows and \(7\) columns.

  4. The linear transformation \(L\from \RNrSpc{2}\to \RNrSpc{3}\) represented by the matrix

    \[ A = \left[ \begin{array}{rr} 4 & 1 \\ 0 & 3 \\ 1 & 2 \end{array} \right] \]

    has domain \(\RNrSpc{2}\) and target \(\RNrSpc{3}\).

  5. The linear transformation \(L\from \RNrSpc{2}\to \RNrSpc{3}\) represented by the matrix

    \[ A = \left[ \begin{array}{rr} 4 & 1 \\ 0 & 3 \\ 1 & 2 \end{array} \right] \]

    has domain \(\RNrSpc{3}\) and target \(\RNrSpc{2}\).

  6. The linear transformation \(L\from \RNrSpc{2}\to \RNrSpc{3}\) represented by the matrix

    \[ A = \left[ \begin{array}{rr} 4 & 1 \\ 0 & 3 \\ 1 & 2 \end{array} \right] \]

    transforms the vector \(\Vect{a}\DefEq (1,-1)\) into the vector \((3,-3,-1)\).

Problem 2

For each of the statements below indicate if it is true or false.

  1. If \(L\from \RNrSpc{2}\to \RNrSpc{2}\) is represented by the matrix

    \[ A = \left[ \begin{array}{rr} 4 & -1 \\ 2 & 6 \end{array} \right] \]

    Then \(L(1,0) = (4,2)\)

  2. If \(L\from \RNrSpc{2}\to \RNrSpc{2}\) is represented by the matrix

    \[ A = \left[ \begin{array}{rr} 4 & -1 \\ 2 & 6 \end{array} \right] \]

    Then \(L(1,0) = (4,-1)\)

  3. If \(L\from \RNrSpc{2}\to \RNrSpc{2}\) is represented by the matrix

    \[ A = \left[ \begin{array}{rr} 4 & -1 \\ 2 & 6 \end{array} \right] \]

    Then \(L(1,0) = (-1,6)\)

  4. If \(L\from \RNrSpc{2}\to \RNrSpc{2}\) is represented by the matrix

    \[ A = \left[ \begin{array}{rr} 4 & -1 \\ 2 & 6 \end{array} \right] \]

    Then \(L(1,1) = (3,8)\)

  5. If \(L\from \RNrSpc{2}\to \RNrSpc{2}\) is represented by the matrix

    \[ A = \left[ \begin{array}{rr} 4 & -1 \\ 2 & 6 \end{array} \right] \]

    Then \(L(0,1) = (-1,6)\)

Problem 3

For the matrices \(\Mtrx{A}\) below find the coordinate description of the corresponding linear map.

  1. \[ A = \left[\begin{array}{rrr} -2 & 1 & 0 \end{array}\right] \]
  2. \[ A = \left[\begin{array}{r} 3 \\ -1 \\ \pi \\ 4 \end{array}\right] \]
  3. \[ A = \left[\begin{array}{rr} a & b \\ c & d \end{array}\right] \]
  4. \[ \Mtrx{A} = \left[\begin{array}{rrr} 2 & 1 & -1 \\ 1 & 2 & -3 \\ 1 & 0 & 1 \end{array}\right] \]
  5. \[ A = \left[\begin{array}{rrrr} 9 & 8 & -6 & -5 \\ -7 & -3 & 4e & 17 \end{array}\right] \]
  6. \[ A = \left[\begin{array}{rrrr} 4 & 0 & 1 & 7 \\ 6 & 0 & 9 & -2 \\ -3 & 0 & 2 & 2 \end{array}\right] \]
  7. \[ A = \left[\begin{array}{rr} 5 & 1 \\ 0 & 0 \\ -9 & 3 \\ -8 & -7 \end{array}\right] \]