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