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\centerline{\large Some Practice Problems for Exam 2 -- Math 221}
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The second exam will cover sections 2.4, 4.1,4.2,4.3,5.1,5.2,5.3,
and 5.4.  The following problems were taken from last year exams.
If anything there are not enough problems from sections 5.3 and
5.4 listed here -- do the ones in the book which were not
assigned.  The problems are not listed in order.

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1. Find all eigenvalues and eigenvectors

$$
\left(\begin{array}{cc}
    1   &  2   \\
    -1 & 1
\end{array}\right)
$$

2.  Find $x_2$ by {\bf Cramer's} rule (no credit will be given if
other method is used).
$$
\begin{array}[t]{lclclcr}
x_1 &   &       &   +     &x_3 & = & 3 \\
x_1 & - &  x_2  &   -     & x_3 & = & -1 \\
    &   &  x_2  &   +     & x_3 & = & 4 
\end{array}
$$

3.  Write down the cofactor expansion along the {\em second} row.
Do not evaluate.

 $$
\left|\begin{array}{ccc}
1   &  2 &   3      \\
4   &  5 &   6      \\
7   &  8 &   0   
\end{array}\right|
$$ 


5. Multiple choice questions.

(i).  Let $\bf u, v$ be two vectors in $R^n$.  If $\|{\bf u}\|=2$
and $\|{\bf w}\| =3$, then

(A). $|{\bf u\cdot w} |\leq 6$, (B). $|{\bf u\cdot w} |= 6$, (C).
$|{\bf u\cdot w} |\geq 6$, (D). none of the above.

(ii). Suppose that $A$ is a $n\times n$ matrix such that det$A =
0$. Let $T_A: R^n\rightarrow R^n$ be a linear operator with its
standard matrix $A$.  Which of the following statements is true?

(A). The range of $T_A$ is $R^n$,  (B). $T_A$ is one to one, 
(C). $T_A$ is invertible, 
(D). none of the above.

7. Determine which sets are vector spaces under the given
operations.

(i). The set of all vectors of the form $(x,y,1)$ with usual
addition and scalar multiplication.

(ii). The set of all $2\times 2$ matrices of the form $
\left(\begin{array}{cc}
    a  &  b   \\
    c & 0
\end{array}\right)
$ with matrix addition and scalar multiplication.

(iii). The set of all real positive functions with usual addition
and scalar multiplication.


(iv).  The set of all real-valued functions $f$ defined everywhere
on the real line and such that $f(1) = 2$, with usual addition
and scalar multiplication.  

8.  Find two vectors in $R^2$ with Euclidean norm 1 whose
Euclidean inner product with $(3, -1)$ is zero.


9. Find the standard matrix for the linear operator defined by
the formula $T(x_1, x_2, x_3) = (x_1 + 2x_2+3x_3, 4x_1-5x_2)$.

10.  Find the standard matrix for the stated composition on
$R^2$:

(a). A counterclockwise rotation by $30^\circ$, followed by a
reflection about the line $x=y$.

(b). An orthogonal projection on the $x$-axis, followed by a
contraction with factor $k=\frac12$.

11. Let $T:R^2\rightarrow R^2$ be a linear operator given by the
formula $T(x_1, x_2) = (x_1 - x_2, x_1 + 2x_2)$.  Find a formula
for the inverse operator $T^{-1}(w_1,w_2)$.

12. Determine which of the following sets of vectors are linearly
dependent.

(a), $x$, $x^2$, $x^3$. 

(b), $(0,0,0)$, $(1,1,1)$, $(-1,0,3)$.

(c), $\cos^2 x$, $\sin^2 x$.  

(d), $(0,0,0,1)$, $(0,0,1,-1)$, $(0, 1,2,3)$. 

(f), $(9,-1,4,2,5)$, $(8,-3,9,-7,4)$, $(3,-3,-6, 4,2)$,
$(3,5,7,-4,8)$, $(3,2,8,9,1)$, $(5,2,8,1,9)$. 

13. Prove that for any vectors ${\bf u, v, w, z}$, the vectors
${\bf u-v}$, ${\bf v-w}$, ${\bf w-z}$, ${\bf z-u}$ form a
linearly dependent set.


14.  The dimension of the space $P_8$ is ($P_8$ is the space of
polynomials of degree 8 or less).

(A). 1, (B). 7,   (C). 8,  (D). 9,  (E). infinity.

15. Prove that every set with 5 vectors or more in $P_3$ is
linearly dependent. ($P_3$ is the space of polynomials of degree
3 or less). 

16.  $T$ is a linear transformation such that $T(\vec{u})=
(2,1,3)$ and $T(\vec{v})= (1,1,1)$.  Compute
$T(2\vec{v}-3\vec{u})$.

17. Consider the vector space $P$ of polynomial functions.  Is
$2x-x^2$ in span of $1-x, 3x-x^2$? Is $3$ in the span of $1-x,
3x-x^2$?

18. Let $T$ be the linear transformation with standard matrix
 $ \begin{bmatrix}
   4 & 0 \\
   1 & 3 \\
  \end{bmatrix} $.  Decide which of the vectors $\vec{u}$ and $\vec{v}$ are eigenvectors
  for $T$.  For the eigenvectors that you find, indicate the
  corresponding eigenvalue.

$$
 \vec{u} = (1,2) \qquad \vec{v} = (0,3)
 $$
 
 19.  Find a basis for the plane in $\mathbb{R}^3$ defined by the 

equation $2 x + y = 0$.

20.  Assume that $\vec{u}$ and $\vec{v}$ are non-$0$ vectors in $\mathbb{R}^2$.
Assume that $\vec{u}$ and $\vec{v}$ are orthogonal.  Show that 
they are linearly independent.
  


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