We are given the system $Ax = b$, where:

$A = \begin{bmatrix} -1 & 3 \\ -4 & 12 \end{bmatrix}, \quad b = \begin{bmatrix} 1 \\ -1 \end{bmatrix}$ A basis for the null space of $A$ is given by: $\begin{bmatrix} 3 \\ 1 \end{bmatrix}$ and a particular solution to $Ax = b$ is: $x_p = \begin{bmatrix} -1 \\ 0 \end{bmatrix}$

We need to find the row space solution $x_r$ for the equation $Ax = b$.

Solution

The general solution for the equation $Ax = b$ can be written as: $x = x_p + x_n$ where $x_p$ is a particular solution to the system, and $x_n$ is any solution in the null space of $A$.

Given that a basis for the null space is $\begin{bmatrix} 3 \\ 1 \end{bmatrix}$, we can express $x_n$ as: $x_n = k \begin{bmatrix} 3 \\ 1 \end{bmatrix}$ for some scalar $k$.

The row space solution $x_r$ represents the component of $x$ that lies in the row space of $A$. Since $x_p = \begin{bmatrix} -1 \\ 0 \end{bmatrix}$ is already a solution to the system, and it does not lie in the null space of $A$, it is the row space solution.

Therefore, the row space solution $x_r$ is: $x_r = \begin{bmatrix} -1 \\ 0 \end{bmatrix}$

Would you like more details on this explanation or have any questions?

Related Questions

1. What is the definition of the row space of a matrix?
2. How do you find a particular solution for a linear system $Ax = b$?
3. What are the steps to find the null space of a matrix?
4. Why is the null space important when solving linear equations?
5. How can we verify if a given vector is a solution to the equation $Ax = b$?

Tip

When solving a system of linear equations, it's helpful to check both the particular solution and the null space solution to ensure the completeness of the general solution.