MA 2071 A '02 Practice Exam 2 Name: .

Section Number:.

  1. First, find an equation for the plane in $\mathbf{R}^3$ containing the origin and perpendicular to the vector $\mathbf{n} =
(1,2,-1)$. Let $W$ be the set of all points in $\mathbf{R}^3$ that lie in this plane. Show that $W$ is a subspace of $\mathbf{R}^3$.

  2. Let $V$ be the set of all ordered pairs, $(x,y)$. Define the $\odot$ operation by $c \odot (x,y) = (cx,-cy)$ Show that this definition does not satisfy property (h) given below.
    Property (h) $1 \odot \mathbf{u} = \mathbf{u}$.

  3. Suppose that $L$ is a linear transformation of $\mathbf{R}^2$ into $\mathbf{R}^2$. Given the following information,

    \begin{displaymath}L(\mathbf{e}_1) = \left[ \begin{array}{c}
1 \\
2
\end{arr...
...}_2) = \left[ \begin{array}{c}
-2 \\
0
\end{array} \right]
\end{displaymath}

    compute $L(\mathbf{u})$, where

    \begin{displaymath}\mathbf{u} = \left[ \begin{array}{c}
-2 \\
4
\end{array} \right]
\end{displaymath}

  4. Find a basis for the subspace of $\mathbf{R}^4$ given by all vectors of the form $(a,b,c,d)$ with $a-c=0$ and $b+c+d=0$. What is the dimension of the subspace?

  5. Let $W$ be the subset of $P_2$ consisting of linear combinations of the three polynomials $\{t-1, t^2, 3t-3\}$. First, show that $W$ is a subspace, and then determine a basis for and the dimension of $W$.

  6. Suppose $\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_k$ are vectors in a vector space $V$. Explain what it means for these vectors to be linearly independent.

  7. Find a basis for the null space of the matrix $A$ given below. What is the dimension of the null space?

    \begin{displaymath}A = \left[ \begin{array}{cccc}
1 & 2 & -4 & 3 \\
0 & 1 & -...
...\
2 & -1 & 2 & 6 \\
-1 & 3 & -6 & -3
\end{array} \right]
\end{displaymath}

William W. Farr
2002-09-23