Suppose 
A
 has n rows and m columns. A linear system of the form 
A
x
→
=
0
→
 is called 'homogeneous' as the constant vector on the right of the equation is the zero vector with n components. Such a system is never inconsistent, since it always have the trivial solution 
x
→
=
0
→
, the zero vector with m components. Any other solution, if there is one, is called a 'nontrivial solution'. Given

A
=
[
3	−9	1	−3	−3	9 
]
,
, find one nontrivial solution of 
A
x
→
=
0
→
 by inspection. Hint: Think of the matrix product in terms of linear combinations of columns. 


x
→
=
 
[
0	1 

Given that the matrix \( A \) is:

\[
A = \begin{bmatrix}
3 & -9 & 1 & -3 & -3 & 9
\end{bmatrix}
\]

We are tasked with finding a nontrivial solution to the homogeneous system \( A \vec{x} = \vec{0} \), where \( \vec{x} \) is a vector with six components (since \( A \) has six columns). The system is never inconsistent because the zero vector is always a solution, but we are looking for a nontrivial solution, meaning a vector \( \vec{x} \neq \vec{0} \).

### Strategy
The matrix product \( A \vec{x} = 0 \) represents a linear combination of the columns of \( A \) being set equal to zero. That is:

\[
3x_1 - 9x_2 + x_3 - 3x_4 - 3x_5 + 9x_6 = 0
\]

We can use the hint provided, which suggests that one component of \( \vec{x} \) is 0, and we should find a vector that satisfies the equation by inspection.

### Step-by-step inspection

By trial, let's set:
\[
\vec{x} = \begin{bmatrix}
0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 1
\end{bmatrix}
\]

Now, check if this vector satisfies the equation \( A \vec{x} = \vec{0} \).

Substitute these values into the equation:

\[
3(0) - 9(1) + 1(1) - 3(0) - 3(0) + 9(1) = 0
\]

Simplifying:

\[
0 - 9 + 1 + 0 + 0 + 9 = 0
\]

This simplifies to:

\[
0 = 0
\]

Thus, the vector \( \vec{x} = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 1 \end{bmatrix} \) is a valid nontrivial solution.

### Conclusion:
The nontrivial solution to the system \( A \vec{x} = 0 \) is:

\[
\vec{x} = \begin{bmatrix} 0 \\ 1 \\ 1 \\ 0 \\ 0 \\ 1 \end{bmatrix}
\]

Would you like more details or have any questions?

---

Here are 5 related questions to expand this topic:
1. How do we determine if a homogeneous system has infinitely many solutions?
2. What conditions must be satisfied for a nontrivial solution to exist in a homogeneous system?
3. How does the rank of matrix \( A \) relate to the number of solutions in the system \( A \vec{x} = \vec{0} \)?
4. What is the geometric interpretation of the solution space for \( A \vec{x} = \vec{0} \)?
5. How can we use row reduction to find nontrivial solutions in more complex systems?

**Tip**: If the number of variables in a system is greater than the number of equations, it's likely that there will be nontrivial solutions.

Suppose A has n rows and m columns. A linear system of the form A*x* = 0 is called 'homogeneous' as the constant vector on the right of the equation is the zero vector with n components. Such a system is never inconsistent, since it always has the trivial solution x = 0, the zero vector with m components. Any other solution, if there is one, is called a 'nontrivial solution'. Given A = [3 −9 1 −3 −3 9], find one nontrivial solution of A*x* = 0 by inspection. Hint: Think of the matrix product in terms of linear combinations of columns.

Explore how to find a nontrivial solution to the homogeneous system A*x* = 0, where A = [3 −9 1 −3 −3 9]. Understand the use of linear combinations of matrix columns in solving homogeneous systems. This problem is ideal for students studying linear algebra at the undergraduate level.

Math Problem Statement

Suppose A has n rows and m columns. A linear system of the form A x →

0 → is called 'homogeneous' as the constant vector on the right of the equation is the zero vector with n components. Such a system is never inconsistent, since it always have the trivial solution x →

A

[ 3 −9 1 −3 −3 9 ] , , find one nontrivial solution of A x →

x →

Solution

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