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Problem 178

# Give an example to illustrate the following theorem: The system of n homogeneous linear equations in $$\mathrm{n}$$ unknowns, $$\mathrm{AX}=0$$, has a nontrivial solution if and only if rank $$\mathrm{A}<\mathrm{n}$$

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Consider a system of 3 homogeneous linear equations in 3 unknowns: $x_1 + 2x_2 + 3x_3 = 0$ $2x_1 + 4x_2 + 6x_3 = 0$ $3x_1 + 6x_2 + 9x_3 = 0$ We represent the system in matrix form AX=0 and row reduce A to find its rank: $\begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix} \sim \begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$ The rank of A is 1, which is less than the number of unknowns (n=3). Since rank(A) < n, there exists a nontrivial solution for the system. For example, the nontrivial solution is $$x_1=-3$$, $$x_2=0$$, and $$x_3=1$$. This illustrates the theorem for a system of 3 homogeneous linear equations in 3 unknowns.
See the step by step solution

## Step 1: Define the system of linear equations

Consider a system of 3 homogeneous linear equations in 3 unknowns as follows: $x_1 + 2x_2 + 3x_3 = 0$ $2x_1 + 4x_2 + 6x_3 = 0$ $3x_1 + 6x_2 + 9x_3 = 0$ #Step 2: Write the system in matrix form#

## Step 2: Write the system in matrix form

We can represent the system of linear equations in the form AX=0, where A is the coefficient matrix and X is the column vector of unknowns: $\begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix} \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix} = 0$ #Step 3: Find the rank of the matrix A#

## Step 3: Find the rank of the matrix A

We determine the rank of the matrix A by row reducing it to its row echelon form: $\begin{pmatrix} 1 & 2 & 3 \\ 2 & 4 & 6 \\ 3 & 6 & 9 \end{pmatrix} \sim \begin{pmatrix} 1 & 2 & 3 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{pmatrix}$ The rank of the matrix A is the number of non-zero rows, which is 1 in this case. #Step 4: Determine if the rank(A) < n#

## Step 4: Determine if the rank(A) < n

Now, we need to compare the rank of A with the number of unknowns (n) in the equation. In this case, n=3, and since rank(A)=1, it means rank(A) < n. #Step 5: Show that there is a nontrivial solution#

## Step 5: Show that there is a nontrivial solution

Since rank(A) < n (1 < 3), there exists a nontrivial solution for the system of linear equations. For example, let's take $$x_3 = 1$$, then we would have: $x_1 = -3x_3 = -3$ $x_2 = (4x_1 + 6x_3)/2 = (4*(-3) + 6)/2 = 0$ The nontrivial solution is $$x_1=-3$$, $$x_2=0$$, and $$x_3=1$$. This shows that the given theorem holds true for the provided example: The system of 3 homogeneous linear equations in 3 unknowns, AX=0, has a nontrivial solution because the rank of A is less than 3 (number of unknowns).

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