Problems

Linear Maps and Linear Equations 1

  1. Determine if the following functions are linear. If they are, write them as a matrix multiplication. If they aren’t, explain why not:

    1. \(T : \mathbb{R}^2 \to \mathbb{R}^3\) defined by \[T\left(\begin{bmatrix} a \\ b \end{bmatrix}\right) = \begin{bmatrix} 2a + b - 1 \\ b + 8a \\ 2a - 3b \end{bmatrix}\]

    2. \(T : \mathbb{R}^3 \to \mathbb{R}^2\) defined by \[T\left(\begin{bmatrix} a \\ b \\ c \end{bmatrix}\right) = \begin{bmatrix} 12a - 2b + 3c \\ 3c - 4b + a \end{bmatrix}\]

    3. \(T : \mathbb{R}^2 \to \mathbb{R}^2\) defined by \[T\left(\begin{bmatrix} a \\ b \end{bmatrix}\right) = \begin{bmatrix} a^2 - b^2 \\ a + b \end{bmatrix}\]

  2. Use Gaussian elimination by hand to solve \[\begin{bmatrix} 1 & -2 & -1 &1 \\ 2 & 1 & 3 &-1 \\ 3 & 3 & 6 & -2 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix} = \begin{bmatrix} 5 \\ -1 \\ -4 \end{bmatrix}\] and then determine if \(\begin{bmatrix} 5 \\ -1 \\ -4 \end{bmatrix}\) can be written in terms of the columns of the matrix.

  3. Use Gaussian elimination by hand to solve \[\begin{bmatrix} 2 & 3 & 1 \\ 4 & 6 & -1 \\ 2 & 3 & 1 \end{bmatrix} \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}\] and then detrmine if \(\begin{bmatrix} 2 \\ 1 \\ 1 \end{bmatrix}\) can be written in terms of the columns of the matrix.

  4. Let \(T : \mathbb{R}^2 \to \mathbb{R}^2\) be a linear map that satisfies \[T\left(\begin{bmatrix} 1 \\ 0 \end{bmatrix}\right) = \begin{bmatrix} 2 \\ 5 \end{bmatrix}\] and \[T\left(\begin{bmatrix} 0 \\ 1 \end{bmatrix}\right) = \begin{bmatrix} -1 \\ 2 \end{bmatrix}\]

    1. Find \(T\left(\begin{bmatrix} 3 \\ -4 \end{bmatrix}\right)\)

    2. Find \(T\left(\begin{bmatrix} a \\ b \end{bmatrix}\right)\) (i.e., find a formula of \(T\))

    3. Find a matrix \(A\) such that \[T\left(\begin{bmatrix} a \\ b \end{bmatrix}\right) = A\begin{bmatrix} a \\ b \end{bmatrix}\]

  5. Let \(T : \mathbb{R}^2 \to \mathbb{R}^2\) be a linear map that satisfies \[T\left(\begin{bmatrix} 2 \\ 5 \end{bmatrix}\right) = \begin{bmatrix} 1 \\ 0 \end{bmatrix}\] and \[T\left(\begin{bmatrix} -1 \\ 2 \end{bmatrix}\right) = \begin{bmatrix} 0 \\ 1 \end{bmatrix}\]

    1. Find \(T\left(\begin{bmatrix} 1 \\ 0 \end{bmatrix}\right)\)

    2. Find \(T\left(\begin{bmatrix} 0 \\ 1 \end{bmatrix}\right)\)

    3. Find \(T\left(\begin{bmatrix} a \\ b \end{bmatrix}\right)\) (i.e., find a formula of \(T\))

    4. Find a matrix \(A\) such that \[T\left(\begin{bmatrix} a \\ b \end{bmatrix}\right) = A\begin{bmatrix} a \\ b \end{bmatrix}\]

  6. Find all solutions \[\begin{align*} x + 2y - z &= 4 \\ 2y + 3z &= -2 \\ z &= 1 \end{align*}\]

  7. Find all solutions \[\begin{align*} 2x - 3y + z + 2w &= 5 \\ y - \frac{1}{2}z - 4w &= -3 \\ z + 2w &= -1 \end{align*}\]

  8. Find all solutions \[\begin{align*} x + 3y - 2z + w &= 2 \\ y - 2w &= -1 \\ \end{align*}\]