The Gram-Schmidt process is an orthogonalization procedure that allows you to transform a set of linearly independent vectors into an orthonormal basis. In other words, given a set of linearly independent vectors, the Gram-Schmidt process generates a set of mutually orthogonal unit vectors that span the same subspace.
Here’s the general algorithm for the Gram-Schmidt process:
1. Start with a set of linearly independent vectors $\{\bold{v}_1, \bold{v}_2, \dots, \bold{v}_n\}$.
2. For each vector $\bold{v}_k$ in the set, subtract its projection onto the subspace spanned by the currently processed orthogonal vectors.
3. Normalize the resulting vector.
Mathematically, this procedure can be described as follows:
For a given set of linearly independent vectors $\{\bold{v}_1, \bold{v}_2, \dots, \bold{v}_n\}$, compute the orthonormal basis $\{\bold{u}_1, \bold{u}_2, \dots, \bold{u}_n\}$ as:
1. $\bold{u}_1 = \frac{\bold{v}_1}{\|\bold{v}_1\|}$
For k = 2, …, n:
2. $\bold{w}_k = \bold{v}_k - \sum_{j=1}^{k-1} (\bold{v}_k \cdot \bold{u}_j) \bold{u}_j$
3. $\bold{u}_k = \frac{\bold{w}_k}{\|\bold{w}_k\|}$
Here, the dot (⋅) denotes the inner product of two vectors, and ∥ ⋅ ∥ represents the Euclidean norm.
**Example:**
Let’s apply the Gram-Schmidt process to the following non-orthogonal basis in ℝ2:
$$\bold{v}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad \bold{v}_2 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}$$
Step 1: Normalize $\bold{v}_1$
$$\bold{u}_1 = \frac{\bold{v}_1}{\|\bold{v}_1\|} = \frac{1}{1} \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 1 \\ 0 \end{pmatrix}$$
Step 2: Subtract projections of $\bold{v}_2$ onto $\bold{u}_1$ and normalize the result
$$\bold{w}_2 = \bold{v}_2 - (\bold{v}_2 \cdot \bold{u}_1) \bold{u}_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix} - (\bold{v}_2 \cdot \bold{u}_1) \bold{u}_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix} - (1) \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$
$$\bold{u}_2 = \frac{\bold{w}_2}{\|\bold{w}_2\|} = \frac{1}{\sqrt{0^2 + 1^2}} \begin{pmatrix} 0 \\ 1 \end{pmatrix} = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$
So, the orthonormal basis after applying the Gram-Schmidt process is:
$$\bold{u}_1 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \quad \bold{u}_2 = \begin{pmatrix} 0 \\ 1 \end{pmatrix}$$
As you can see, the Gram-Schmidt process has successfully converted the initial set of vectors into an orthonormal basis.