B) False

2.) If $\phi$ is a solution to a first order linear homogeneous differential equation, then $c\phi$ is also a solution to this equation.

                   A) True                    

3.) If $\phi$ is a solution to a first order linear homogeneous differential equation with constant coefficients, then $c\phi$ is also a solution to this equation.

                   A) True                    

1.) If $b^2 - 4ac < 0$ then the solution to the initial value problem $ay'' + by' + cy = 0$, $y(0) = -1$, $y'(0) = -3$ is complex valued.

                                        B) False

2.) If $b^2 - 4ac < 0$ then the solution to the initial value problem $ay'' + by' + cy = 0$, $y(0) = -1$, $y'(0) = -3$ is real valued.

                   A) True                    

1.) $L(g) = g'' + p(t) g' + q(t)g$ is a linear function on the space of all twice differentiable functions.

                   A) True            

2.) $L(g) = g'' + p(t) g' + q(t)gg'$ is a linear function on the space of all twice differentiable functions.

                                B) False

1.) There is a unique solution to the differential equation ay'' + by + cy = g(t), y(0) = 1, y'(0) = 3

                   A) True            

2.) There is a unique solution to the differential equation ay'' + by + cy = g(t), y(0) = 1, y(1) = 3

                                B) False

3.) There is a unique solution to the differential equation ay'' + by + cy = g(t), y(0) = 1, y'(1) = 3

                                B) False

4.) There is a unique solution to the differential equation ay'' + by + cy = g(t), y(1) = 1, y'(1) = 3

                   A) True            

1.) When taking the derivative with respect to $t$, $(y^2)' = 2y$

                                B) False

2.) When taking the derivative with respect to $t$, $(y^2)' = 2yy'$

                   A) True            

3.) If $y(0) = -2$ and $y^2 = g(t)$, then $y(t) = \sqrt{g(t)}$

                                B) False

4.) If $y(0) = -2$ and $y^2 = g(t)$, then $y(t) = -\sqrt{g(t)}$

                   A) True            

1. If $p:(a, b) \rightarrow R$, $q:(a, b) \rightarrow R$, and $g:(a, b) \rightarrow R$ are continuous and $a < t_0 < b$, then there exists a unique function $y = \phi(t)$,   $\phi:(a, b) \rightarrow R$ that satisfies the initial value problem $y'' + p(t) y' + q(t)y = g(t)$,   $y(t_0) = y_0$,    $y'(t_0) = y_1$.

               A) True                        

2. $D(f) = f'$ is a linear function.

               A) True                        

3. The differential operator $D$ is a linear operator

               A) True                        

               A) True                        

1. Suppose $A \left[\matrix{5 \cr 6}\right] = \left[\matrix{5 \cr 13}\right]$,   $A \left[\matrix{3 \cr 5}\right] = \left[\matrix{9 \cr 15}\right]$,   $A \left[\matrix{-1 \cr 3}\right] = \left[\matrix{17 \cr 19}\right]$,   $A \left[\matrix{2 \cr 1}\right] = \left[\matrix{-4 \cr -2}\right]$

State the 2 eigenvalues of $A$:    3, -2

State 5 eigenvectors of $A$: $\left[\matrix{3 \cr 5}\right]$,   $ \left[\matrix{6 \cr 10}\right]$,   $ \left[\matrix{9 \cr 15}\right]$,   $ \left[\matrix{2 \cr 1}\right]$,   $ \left[\matrix{4 \cr 2}\right]$,  ...

Observation:
$A \left[\matrix{5 \cr 6}\right]$ $= A \left( \left[\matrix{3 \cr 5}\right] + \left[\matrix{2 \cr 1}\right]\right)$ $= A \left[\matrix{3 \cr 5}\right] + A \left[\matrix{2 \cr 1}\right]$ $= 3 \left[\matrix{3 \cr 5}\right] - 2 \left[\matrix{2 \cr 1}\right]$ $= \left[\matrix{5 \cr 13}\right]$

$A \left[\matrix{-1 \cr 3}\right]$ $= A \left( \left[\matrix{3 \cr 5}\right] - 2 \left[\matrix{2 \cr 1}\right]\right)$ $= A \left[\matrix{3 \cr 5}\right] -2 A \left[\matrix{2 \cr 1}\right]$ $= 3 \left[\matrix{3 \cr 5}\right] + 4 \left[\matrix{2 \cr 1}\right]$ $= \left[\matrix{17 \cr 19}\right]$

3.) $ \left[\matrix{5 \cr 6}\right], \left[\matrix{4 \cr 13}\right]$ are linearly independent.

               A) True                        

4.) $ \left[\matrix{5 \cr 6}\right], \left[\matrix{4 \cr 13}\right], \left[\matrix{1 \cr 1}\right]$ are linearly independent.

                                        B) False

5.) $ \left[\matrix{5 \cr 6}\right], \left[\matrix{-10 \cr -12}\right]$ are linearly independent.

                                        B) False

The following are all equivalent:

$v_1, v_2, ..., v_n$ are linearly independent

$c_1v_1 + c_2v_2 + ... + c_nv_n = 0$ has a unique of solutions

There does not exists an $i$ such that $v_i$ is a linear combination of $v_1, ..., v_{i-1}, v_{i+1}, ..., v_n$.

Let $A = [v_1 v_2 ... v_n]$

$Ax = 0$ has a unique solution.

$Ax = b$ has at most one solution.

If $A$ is a square matrix, $Ax = b$ has a unique solution.

If $A$ is a square matrix, $det(A) \not= 0$

0 is not an eigenvalue of $A$.

1. If $p:(a, b) \rightarrow R$, $q:(a, b) \rightarrow R$, and $g:(a, b) \rightarrow R$ are continuous and $a < t_0 < b$, then there exists a unique function $y = \phi(t)$,   $\phi:(a, b) \rightarrow R$ that satisfies the initial value problem $y'' + p(t) y' + q(t)y = g(t)$,   $y(t_0) = y_0$,    $y'(t_0) = y_1$.

               A) True                        

2. If $A$ is a matrix whose elements consist of continuous functions of $t$, then there exists a unique function ${\bf x}(t) = {\bf f}(t)$ that satisfies the initial value problem ${\bf x}' = A{\bf x}$ ,   ${\bf x}(t_0) = {\bf x_0}$,    ${\bf x}'(t_0) = {\bf x_1}$.

                                        B) False

3. If $A$ is a matrix whose elements consist of continuous functions of $t$, then there exists a unique function ${\bf x}(t) = {\bf f}(t)$ that satisfies the initial value problem ${\bf x}' = A{\bf x}$ ,   ${\bf x}(t_0) = {\bf x_0}$.

               A) True