Exercises - Vectors and Linear Transformations

  1. Suppose $M$ is a linear transformation operating on 2-dimensional vectors.

    If $M \begin{pmatrix}1\\0\end{pmatrix} = \begin{pmatrix}4\\5\end{pmatrix}$ and $M \begin{pmatrix}0\\1\end{pmatrix} = \begin{pmatrix}-3\\7\end{pmatrix}$, find $M \begin{pmatrix}-6\\8\end{pmatrix}$  

  2. We know that we can find the matrix form for a linear transformation, $T$, on two-dimensional vectors, if we know the output of $T\begin{pmatrix}1\\0\end{pmatrix}$ and $T\begin{pmatrix}0\\1\end{pmatrix}$. What if we know the output of $T$ when applied to other vectors? Can we still find the matrix form of $T$? Find the matrix form for $T$ if the below facts are known:

    $$T \begin{pmatrix}2\\0\end{pmatrix} = \begin{pmatrix}14\\20\end{pmatrix} \quad \quad \textrm{and} \quad \quad T \begin{pmatrix}3\\1\end{pmatrix} = \begin{pmatrix}10\\11\end{pmatrix}$$

  3. Show that if $F$ and $G$ are linear transformations that operate on two-dimensional vectors, then the matrix representation of $F-G$ is given by the following:

    $$\begin{bmatrix}a & b\\c & d\end{bmatrix}- \begin{bmatrix}e & f\\g & h\end{bmatrix} = \begin{bmatrix}a-e & b-f\\c-g & d-h\end{bmatrix}$$

  4. Suppose $F$ is a linear transformation that operates on three-dimensional vectors. Let us adopt the convention of writing such transformations $F$ as matrices in the following form

    $$F = \left[ \begin{array}{ccc}
    a & d & g\\
    b & e & h\\
    c & f & i
    \end{array} \right]$$

    to imply that

    $$F \begin{pmatrix}1\\0\\0\end{pmatrix} = \begin{pmatrix}a\\b\\c\end{pmatrix}, \quad F \begin{pmatrix}0\\1\\0\end{pmatrix} = \begin{pmatrix}d\\e\\f\end{pmatrix}, \quad \textrm{ and } \quad F \begin{pmatrix}0\\0\\1\end{pmatrix} = \begin{pmatrix}g\\h\\i\end{pmatrix}$$

    1. Use the linearity of $F$ to determine $F\begin{pmatrix}x\\y\\z\end{pmatrix}$.

    2. If $F$ and $G$ are the following linear transformations that operate on three-dimensional vectors, find the matrix representation of their sum and composition.

      $$F = \left[ \begin{array}{ccc}
      a & d & g\\
      b & e & h\\
      c & f & i
      \end{array} \right] \quad, \quad G = \left[ \begin{array}{ccc}
      j & m & p\\
      k & n & q\\
      l & o & r
      \end{array} \right]$$

    3. Use the matrix representations found above to find the matrix representation of the following two linear transformations.

      $$\left[ \begin{array}{ccc}
      1 & 2 & 3\\
      4 & 5 & 6\\
      7 & 8 & 9
      \end{array} \right] +
      \left[ \begin{array}{ccc}
      1 & -1 & 2\\
      2 & -3 & 5\\
      -1 & 2 & 7
      \end{array} \right]$$
      $$\left[ \begin{array}{ccc}
      1 & 2 & 3\\
      4 & 5 & 6\\
      7 & 8 & 9
      \end{array} \right] \circ
      \left[ \begin{array}{ccc}
      1 & -1 & 2\\
      2 & -3 & 5\\
      -1 & 2 & 7
      \end{array} \right]$$

  5. Suppose $F$ rotates two-dimensional vectors by $\theta$ degrees, counter-clockwise about the origin. Convince yourself that $F$ is a linear transformation, and then find its matrix representation. Finally, use this matrix to rotate two vectors of your choosing by $25^{\circ}$.  

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