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 $25^{\circ}$.

◆ ◆ ◆