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When we talk about an **orthogonal basis**, we refer to a set of vectors in a vector space where each pair of different vectors is orthogonal to each other. This simply means that the dot product of any two different vectors from the set is zero, indicating right angles between them. To convert a given set of vectors into an orthogonal basis, we use the **Gram-Schmidt process**.

• Start with the original vectors and choose the first vector of your basis from them.
• For each subsequent vector, subtract the component in the direction of every vector already chosen for your basis.

The vectors **\( \mathbf{x}_1 = \begin{bmatrix} 1 & 1 \end{bmatrix} \)** and \( \mathbf{x}_2 = \begin{bmatrix} 1 & 2 \end{bmatrix} \)** from our example were transformed into an orthogonal set through projection: \[ \mathbf{v}_2 = \mathbf{x}_2 - \text{proj}_{\mathbf{v}_1}(\mathbf{x}_2) \] This ensures the orthogonality of the resulting vectors.

An **orthonormal basis** takes the concept of an orthogonal basis one step further. Yes, the vectors are orthogonal, but they are also of unit length. In simple terms, each vector in an orthonormal set has a length or magnitude of one. This is particularly useful because calculations with these vectors are typically simpler and more stable.

• To normalize a vector, divide it by its own magnitude.
• The magnitude of a vector is calculated using the square root of the sum of the squares of its components.

For example, from our orthogonal vectors \( \mathbf{v}_1 = \begin{bmatrix} 1 & 1 \end{bmatrix} \) and \( \mathbf{v}_2 = \begin{bmatrix} -\frac{1}{2} & \frac{1}{2} \end{bmatrix} \), we normalized them to form the **orthonormal basis**: \[ \mathbf{u}_1 = \frac{1}{\sqrt{2}} \begin{bmatrix} 1 & 1 \end{bmatrix} \] and \[ \mathbf{u}_2 = \begin{bmatrix} -\frac{1}{\sqrt{2}} & \frac{1}{\sqrt{2}} \end{bmatrix} \]. This ensures not only orthogonality but also a unit norm for each vector.

Understanding the **dimension of a vector space** is foundational in linear algebra. It essentially tells us the number of vectors in a basis for the space, which is the smallest set of vectors needed to span the space. In our example, we're dealing with \( \mathbb{R}^{2} \), so the dimension is 2. This means that any vector in \( \mathbb{R}^{2} \) can be expressed as a combination of just two independent vectors.The concept of dimension is rooted in the idea of how many directions are available in the space you're working with. For \( \mathbb{R}^{2} \), those directions are typically interpreted as the conventional x and y axes.

• A vector space can have various types of sets of vectors, with a basis being the most efficient one.
• All bases for a given space will have the same number of vectors, equal to the dimension.

Thus, in our exercise, the vectors \( \mathbf{x}_1 \) and \( \mathbf{x}_2 \) serve as a basis for \( \mathbb{R}^{2} \), echoing the space's two-dimensional nature.