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A homogeneous system of linear equations is a system in which every equation has zero as its constant term. In simple terms, it can be described as:\[ A\mathbf{x} = \mathbf{0} \]where \(A\) is a matrix representing the coefficients and \(\mathbf{x}\) is the vector of variables. The homogeneous system always has at least one solution: the trivial solution, which is when all variables equal zero.

However, it's often more interesting to find the non-trivial solutions, if they exist. These solutions form a vector space known as the null space or kernel of the matrix \(A\). In the context of our example, this system is a single equation in three variables: \(x_1 - 2x_2 + x_3 = 0\).

To find the solution, we express our variables with respect to a free variable. For example, we can choose \(x_2\) as our free variable and express \( x_1 = 2x_2 \) and \(x_3 = -2x_2 \) as described in the solution. This highlights how the solutions form a one-dimensional subspace, essentially a line through the origin in three-dimensional (3D) space.

The Gram-Schmidt process is a method used to convert a set of vectors into an orthogonal set. This orthogonal set forms the basis of a subspace and can be further normalized into an orthonormal basis.

This process is particularly useful when working with vector spaces where orthogonality (vectors being at right angles) simplifies computations substantially.

• In our example, we were given a single vector \((2, 1, 2)\) to represent the solution space.
• A single vector is trivially orthogonal since there are no other vectors to form an orthogonal pair or set.

Thus, in this case, the Gram-Schmidt process doesn't add much as the single vector is already orthogonal by definition. However, remembering that this process is crucial when dealing with multiple vectors in a basis can be helpful for more complex problems.

Normalization is the process by which a vector is converted into a unit vector. A unit vector has a length or magnitude of 1. This is important because it maintains the direction of the vector but simplifies its use in calculations by standardizing its length.

The steps to normalize a vector involve dividing each component of the vector by its magnitude. The magnitude \(||v||\) is calculated using the following formula for a vector \((x, y, z)\): \[|| \mathbf{v} ||= \sqrt{x^2 + y^2 + z^2} \]

• Applying this to our vector \((2, 1, 2)\), we calculate \(||\mathbf{v}|| = \sqrt{4 + 1 + 4} = 3 \).
• Thus, the normalized vector becomes \( \left( \frac{2}{3}, \frac{1}{3}, \frac{2}{3} \right) \).

The outcome is a vector that lies on the same line as the original but has been scaled down to a unit length, providing an orthonormal basis for the solution space of the given system.