91Ó°ÊÓ

When we talk about a subspace of a vector space, we refer to a smaller space within a larger vector space that itself forms a vector space. For example, in our exercise, we are working with a subspace \( W \) of \( \mathbb{R}^4 \). The subspace is defined by the equations \( x_1 = x_2 + 2x_3 \) and \( x_4 = -x_2 + x_3 \).

This means that every vector in \( W \) is constructed based on this relationship among the components of the vectors. A subspace will always include the origin vector (the zero vector) and allows linear combinations of vectors within it, following two main rules:

• Closure under addition: The sum of any two vectors in the subspace is also in the subspace.
• Closure under scalar multiplication: If a vector is in the subspace, any scalar multiple of that vector is also in the subspace.

These properties ensure that \( W \) is a subspace of the entire vector space \( \mathbb{R}^4 \).

The Gram-Schmidt process is a technique used to convert a set of vectors into an orthonormal set, meaning that all vectors are orthogonal (perpendicular) to each other, and each is a unit vector (of length 1).

In the context of this exercise, the goal of applying the Gram-Schmidt process is to transform the basis vectors \( [1, 1, 0, -1] \) and \( [2, 0, 1, 1] \) into an orthonormal basis. The process involves taking each vector in turn, subtracting the projection of the vector onto all previous orthonormal vectors, and then normalizing the result:

• Start with the first vector, \( v_1 = [1, 1, 0, -1] \), and find the unit vector by dividing by its magnitude.
• Proceed with the second vector \( v_2 = [2, 0, 1, 1] \), subtract its projection onto \( u_1 \) (the orthonormal version of \( v_1 \)), and normalize the result to get \( u_2 \).

This step-by-step correction and normalization ensures that the vectors in the resulting set are both orthogonal and normal, meaning they form an orthonormal basis.

Linear combinations are fundamental in expressing vectors in a subspace. For our exercise, a vector in the subspace \( W \) of \( \mathbb{R}^4 \) can be expressed as a linear combination of the vectors \( [1, 1, 0, -1] \) and \( [2, 0, 1, 1] \).

Understanding this means we can write any vector in \( W \) as:

• \( t[1, 1, 0, -1] + s[2, 0, 1, 1] \) where \( t \) and \( s \) are scalars.

This representation simplifies working with the subspace, as it allows us to leverage the basis vectors to describe an infinite number of vectors in \( W \) where \( t \) and \( s \) can be any real numbers. The versatility of linear combinations lies at the heart of many concepts in linear algebra, such as solving systems of equations and understanding span and independence.

Normalization is the process of converting a vector into a unit vector or a vector with a magnitude of 1. In this exercise, we normalize vectors to ensure they are part of an orthonormal basis for our subspace.

To normalize a vector, we divide the vector by its magnitude (length). The magnitude of a vector \( v = [v_1, v_2, v_3, v_4] \) is calculated as \( \|v\| = \sqrt{v_1^2 + v_2^2 + v_3^2 + v_4^2} \). For normalization:

• Calculate the magnitude of the vector.
• Divide each component of the vector by this magnitude.

For example, if \( v = [1, 1, 0, -1] \), we calculate \( \|v\| = \sqrt{1^2 + 1^2 + 0^2 + (-1)^2} = \sqrt{3} \). Then, the normalized vector \( u_1 \) becomes \( \frac{1}{\sqrt{3}}[1, 1, 0, -1] \).
Normalization is crucial for obtaining orthonormal bases, ensuring the vectors not only are orthogonal but also have unit length.