1What is the method for solving homogeneous linear systems? (Ax=0)
1. Get the linear system into matrix form Ax = 0
2. Get A into RREF (RREF(A) = A')
3. Multiply out A' x = 0 to get equations
3. Any column in A' without a pivot corresponds to a free variable. i.e., no pivot in column 1 means $x_1$ is free
4. Express non-free variables in terms of free variables.
2What is the augmented matrix [A|B] if $A = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}, \quad B = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$
$\left[\begin{array}{cc|c}
1 & 2 & 1 \\
3 & 4 & 1
\end{array}\right]$
3What is the method for solving inhomogeneous linear systems? (Ax=b)
1. Augment A with b ([A | b])
2. Get [A | b] into RREF (RREF([A | b]) = [A' | b'] )
3. Multiply out A' x = b' to get equations
4. Any column in A' without a pivot corresponds to a free variable i.e. no pivot in column 1 means $x_1$ is free
5. Express non-free variables in terms of free variables.
4How do you apply the RECIPE? e.g. if you have A' x = b' where x = $\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ x_4 \end{bmatrix}$ and b' = $\begin{bmatrix} α \\ β \\ γ \end{bmatrix}$
Take b' and place 0's in the rows with the same index as the free variables e.g. if $x_3$ = free a solution to x is $\begin{bmatrix} α \\ β \\ 0 \\γ \end{bmatrix}$
5How can you write the set of equations
$\begin{aligned}
x_1 &= \frac{5}{4}x_4 + 6 \\
x_2 &= -\frac{3}{2}x_4 - 12 \\
x_3 &= \frac{3}{4}x_4 + 2 \\
x_4 &= \text{free}
\end{aligned}$
as a matrix in the form x = ...?
$x = x_4 \begin{bmatrix} \frac{5}{4} \\ -\frac{3}{2} \\ \frac{3}{4} \\ 1 \end{bmatrix} + \begin{bmatrix} 6 \\ -12 \\ 2 \\ 0 \end{bmatrix}$
6For Ax = b, if RREF(A|b) has a row in the form [0 0 0 ... | β] what can we say about the solutions for x?
x has no solutions
7For Ax = 0, if A is right invertible what can we say about the solutions for x?
x has a non-zero solution
8For Ax = 0, if A is left invertible what can we say about the solutions for x?
x has a unique solution
9For Ax = b, if A is right invertible what can we say about the solutions for x?
x has a solution
10For Ax=b, if A is invertible what can we say about the solutions for x?
x has a unique solution
11What result relates the general solution to Ax=b, the specific soluttion to Ax=b and the general solution to Ax=0
(General solution to Ax=b) = (Specific solition to Ax=b + (General solution Ax=0)
12When finding the RREF of an augmented matrix what can’t you do?
Choose a pivot to be in the augmented (right-hand) side of the matrix