Math Insight

Matrix vector multiplication

Math 2241, Spring 2023
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Due date: Feb. 1, 2023, 11:59 p.m.
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Total points: 1
  1. Vector representation.
    Need help?
    1. Write the vector $\vc{x}=\left ( 1, \quad 2\right )$ as a column matrix. $\vc{x} =$

      Write the vector $\vc{z} = \left[\begin{matrix}- \frac{3}{2}\\2\end{matrix}\right]$ as an ordered pair. $\vc{z} =$

      Need help?
    2. Feedback from applet
      vectors:

      Plot each of the following vectors on the axes to the right. Draw an arrow starting at the origin that extends horizontally an amount equal to the first coordinate and vertically an amount equal to the second coordinate.

      $\left ( 1, \quad 2\right )$, $\left[\begin{matrix}- \frac{3}{2}\\2\end{matrix}\right]$, $\left ( 1, \quad - \frac{1}{2}\right )$, $\left[\begin{matrix}-1\\1\end{matrix}\right]$, $\left ( 0, \quad 1\right )$, $\left ( \frac{3}{2}, \quad 0\right )$, $\left[\begin{matrix}-2\\- \frac{3}{2}\end{matrix}\right]$, $\left ( -2, \quad 0\right )$

      On the graph to the right, increase $n_v$ to reveal vectors. Then, drag the points on the vectors so that each vector above is represented by an arrow. (The color or order of vectors doesn't matter.)

    3. Determine the coordinates of the four vectors drawn in the plot to the right. Enter them in any order. (All coordinates are integers.)


  2. Multiplying a vector or a matrix by a scalar (a real number) just multiples each component by that scalar.
    1. If $\vc{x}=\left ( 3, \quad -1, \quad 5\right )$, then $3\vc{x} = $
      .
    2. If $\vc{x} = \left[\begin{matrix}-3\\-11\\5\end{matrix}\right]$ and $a=-3$, then $a\vc{x} =$
      .
    3. Let $\vc{z} = \left ( a, \quad b\right )$. Calculate $c\vc{z}=$
      .
    4. If $A=\left[\begin{matrix}-3 & 5\\-2 & 7\end{matrix}\right]$ and $c=3$, then $cA=$
    5. If $B=\left[\begin{matrix}4 & -8\\2 & -5\end{matrix}\right]$, then $-4B=$
    6. Feedback from applet
      vectors:
      Let $\vc{u} = \left[\begin{matrix}-1\\-2\end{matrix}\right]$, $\vc{v} = \frac{1}{2}\vc{u}$, and $\vc{w} = - \frac{3}{2}\vc{u}$. Then:
      $\vc{v} = $

      $\vc{w} =$
      .

      On the graph to the right, change $n_v$ to 1 and plot the vector $\vc{u}$ using the blue vector that appears. Then, change $n_v$ to 2 and plot the vector $\vc{v}$ with the green vector that appears. Finally, change $n_v$ to 3 and plot the vector $\vc{w}$ with the red vector that appears.

      Multiplying a vector by the scalar $\frac{1}{2}$ transforms the vector in what way?
      Multiplying a vector by the scalar $- \frac{3}{2}$ transforms the vector in what way?

  3. Matrix-vector multiplication
    1. Recall that a matrix with $m$ rows and $n$ columns can be multiplied by a vector with $n$ components. For each row, we take each entry and multiply by the corresponding component in the vector, then add them up. Doing this for each row yields a vector with $m$ components. For example, consider $$\displaystyle \begin{bmatrix} 4 & 2 & -1 \\ 3 & 2 & 5 \end{bmatrix} \begin{bmatrix} 2 \\ -1 \\ 1 \end{bmatrix}$$ The resulting product is a vector with
      components. The first component is given by

      $\cdot$
      $+$
      $\cdot$
      $+$
      $\cdot$
      $=$

      and the second component is given by

      $\cdot$
      $+$
      $\cdot$
      $+$
      $\cdot$
      $=$

      Note: for the computer to give you credit, make sure the order of the terms matches the order of the entries in the matrix/vector.
    2. $\displaystyle \left[\begin{matrix}4 & 3\\2 & -1\end{matrix}\right]\left[\begin{matrix}3\\-2\end{matrix}\right]=$
    3. $\displaystyle \left[\begin{matrix}5 & 3 & -2\\1 & 4 & 1\\2 & -1 & 3\end{matrix}\right]\left[\begin{matrix}4\\5\\2\end{matrix}\right]=$
    4. $\displaystyle \left[\begin{matrix}-2 & 4 & 2\\3 & 1 & 4\\-1 & 1 & 3\end{matrix}\right]\left[\begin{matrix}2\\-1\\2\end{matrix}\right]=$

  4. Matrix-vector multiplication in two dimensions.
    Need help?
    1. Let $A$ be the matrix $A=\left[\begin{matrix}1 & 2\\3 & 4\end{matrix}\right]$. Let $\vc{x}$ be the vector $\vc{x} = \left[\begin{matrix}5\\6\end{matrix}\right]$.
      The matrix-vector product $A\vc{x} = $
      .
    2. Calculate $B\vc{y}$ where $B=\left[\begin{matrix}-3 & 7\\-2 & 1\end{matrix}\right]$ and $\vc{y} = \left ( 4, \quad 1\right )$.
      $B\vc{y} =$
      Need help?
    3. If $A=\left[\begin{matrix}a & b\\c & d\end{matrix}\right]$ and $\vc{x} = \left[\begin{matrix}x\\y\end{matrix}\right]$, then $A\vc{x} = $
      Need help?
    4. If $I=\left[\begin{matrix}1 & 0\\0 & 1\end{matrix}\right]$ and $\vc{x} = \left[\begin{matrix}x\\y\end{matrix}\right]$, then $I\vc{x} = $

      How did multiplication by the matrix $I$ transform the vector $\vc{x}$?

    5. Feedback from applet
      vectors:
      Let the vector $\vc{v}$ be the product of the matrix $B=\left[\begin{matrix}2 & 0\\0 & 2\end{matrix}\right]$ and the vector $\vc{u}=\left ( 2, \quad -1\right )$.
      Then, $\vc{v} =B\vc{u}=$

      On the graph to the right, change $n_v$ to 1 and plot the vector $\vc{u}$ using the blue vector that appears. Then, change $n_v$ to 2 and plot the vector $\vc{v}$ with the green vector that appears.

      How did multiplication by the matrix $B$ transform the vector $\vc{u}$?

    6. Feedback from applet
      vectors:
      If $A=\left[\begin{matrix}-2 & 4\\1 & -3\end{matrix}\right]$ and $\vc{y}=\left[\begin{matrix}4\\1\end{matrix}\right]$, then
      $\vc{z} = A\vc{y}=$
      .

      On the graph to the right, change $n_v$ to 1 and plot the vector $\vc{y}$ using the blue vector that appears. Then, change $n_v$ to 2 and plot the vector $\vc{z}$ with the green vector that appears.

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Math 2241, Spring 2023