Vectors and matrices Knowledge-Based Download PDF

Questions

1. 1.

   What is the area of the parallelogram spanned by the vectors $\mathbf{u} = \begin{pmatrix} 4 \\ -3 \end{pmatrix}$ and $\mathbf{v} = \begin{pmatrix} 2 \\ 5 \end{pmatrix}$?

   • A) 7
   • B) 14
   • C) 23
   • D) 26
2. 2.

   Given the matrix $A = \begin{pmatrix} 4 & -2 \\ 1 & 3 \end{pmatrix}$ and the vector $\mathbf{w} = \begin{pmatrix} 6 \\ 5 \end{pmatrix}$, what is the vector $\mathbf{v}$ such that $A\mathbf{v} = \mathbf{w}$?

   • A) $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$
   • B) $\begin{pmatrix} 1 \\ 2 \end{pmatrix}$
   • C) $\begin{pmatrix} 2 \\ -1 \end{pmatrix}$
   • D) $\begin{pmatrix} -1 \\ 2 \end{pmatrix}$
3. 3.

   Given the matrix $A = \begin{pmatrix} 3 & 1 \\ 2 & 2 \end{pmatrix}$, which of the following non-zero vectors satisfies the equation $A\mathbf{v} = 4\mathbf{v}$?

   • A) $\begin{pmatrix} 2 \\ -2 \end{pmatrix}$
   • B) $\begin{pmatrix} 1 \\ -2 \end{pmatrix}$
   • C) $\begin{pmatrix} 2 \\ 2 \end{pmatrix}$
   • D) $\begin{pmatrix} 2 \\ 1 \end{pmatrix}$
4. 4.

   Let the matrix $R = \begin{pmatrix} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{pmatrix}$ represent a counterclockwise rotation about the origin by an angle $\theta$. What is the image of the vector $\mathbf{v} = \begin{pmatrix} 2 \\ 0 \end{pmatrix}$ under this transformation when $\theta = \frac{\pi}{3}$?

   • A) $\begin{pmatrix} \sqrt{3} \\ 1 \end{pmatrix}$
   • B) $\begin{pmatrix} 1 \\ -\sqrt{3} \end{pmatrix}$
   • C) $\begin{pmatrix} 1 \\ \sqrt{3} \end{pmatrix}$
   • D) $\begin{pmatrix} -1 \\ \sqrt{3} \end{pmatrix}$
5. 5.

   Given the vector $\mathbf{u} = \begin{pmatrix} 3 \\ -2 \end{pmatrix}$ and the matrix $M = \begin{pmatrix} 1 & 4 \\ -2 & 3 \end{pmatrix}$, what is the value of the dot product $\mathbf{u} \cdot (M\mathbf{u})$?

   • A) $-39$
   • B) $-9$
   • C) $9$
   • D) $21$