The question presented in the image is:

True or False?
If $c_1, c_2, c_3$ are three columns of a $3 \times 3$ matrix $A$, and
$b = 2c_1 - 3c_2 + 5c_3$,
then the linear system $Ax = b$ has at least one solution.

Explanation:

To determine if this statement is true, we need to consider the linear system $A x = b$, where $A$ is a matrix with columns $c_1, c_2, c_3$ and $b$ is a linear combination of those columns. Specifically, $b$ is given as a combination of the columns of $A$:

$b = 2c_1 - 3c_2 + 5c_3.$

This means that $b$ can be expressed as a linear combination of the columns of $A$, implying that the vector $b$ lies within the column space of $A$. Since $b$ is in the column space of $A$, the system $A x = b$ has at least one solution (because $b$ is reachable by some vector $x$).

Conclusion:

The statement is True, because the vector $b$ is a linear combination of the columns of $A$, meaning that $b$ is in the column space of $A$, and thus the system has at least one solution.

Would you like more details or have any further questions?

Here are five related questions:

1. How can we determine the uniqueness of the solution in such a system?
2. What is the column space of a matrix, and how does it relate to the solution of a system?
3. Can a system have more than one solution if the vector $b$ is in the column space of $A$?
4. What is the difference between a homogeneous and non-homogeneous system?
5. How can we verify if a vector $b$ lies in the column space of a matrix?

Tip: A matrix equation $A x = b$ has a solution if and only if $b$ is in the column space of $A$.