Gram-Schmidt Process | Generated by AI
The Gram-Schmidt process is a method for orthonormalizing a set of linearly independent vectors in an inner product space (typically (\mathbb{R}^n) or (\mathbb{C}^n)). It converts a given basis into an orthonormal basis, where the vectors are mutually orthogonal and have unit length.
Algorithm: Gram-Schmidt Process
Given a set of linearly independent vectors ( {v_1, v_2, \dots, v_n} ) in an inner product space, we construct an orthonormal basis ( {u_1, u_2, \dots, u_n} ) as follows:
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Step 1: Compute the first orthonormal vector
[ u_1 = \frac{v_1}{|v_1|} ] -
Step 2: Make the second vector orthogonal to the first and normalize it
[ v_2’ = v_2 - \text{proj}_{u_1}(v_2) = v_2 - \frac{\langle v_2, u_1 \rangle}{\langle u_1, u_1 \rangle} u_1 ] [ u_2 = \frac{v_2’}{|v_2’|} ] -
Step 3: Repeat for the remaining vectors
For ( k = 3, \dots, n ): [ v_k’ = v_k - \sum_{j=1}^{k-1} \frac{\langle v_k, u_j \rangle}{\langle u_j, u_j \rangle} u_j ] [ u_k = \frac{v_k’}{|v_k’|} ]
Here, ( \text{proj}_{u_j}(v_k) = \frac{\langle v_k, u_j \rangle}{\langle u_j, u_j \rangle} u_j ) represents the projection of ( v_k ) onto ( u_j ).
Example: Applying Gram-Schmidt to (\mathbb{R}^3)
Given the vectors:
[ v_1 = (1, 1, 0), \quad v_2 = (1, 0, 1), \quad v_3 = (0, 1, 1) ]
Step 1: Normalize ( v_1 )
[ u_1 = \frac{v_1}{|v_1|} = \frac{(1,1,0)}{\sqrt{2}} = \left(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}}, 0\right) ]
Step 2: Orthogonalize ( v_2 ) against ( u_1 )
[ \text{proj}_{u_1}(v_2) = \frac{\langle v_2, u_1 \rangle}{\langle u_1, u_1 \rangle} u_1 ]
[ = \frac{(1,0,1) \cdot (1/\sqrt{2}, 1/\sqrt{2}, 0)}{1} \cdot (1/\sqrt{2}, 1/\sqrt{2}, 0) ]
[ = \frac{1 \cdot 1/\sqrt{2} + 0 \cdot 1/\sqrt{2} + 1 \cdot 0}{1} \cdot (1/\sqrt{2}, 1/\sqrt{2}, 0) ]
[ = \frac{1/\sqrt{2}}{1} \cdot (1/\sqrt{2}, 1/\sqrt{2}, 0) = \left(\frac{1}{2}, \frac{1}{2}, 0\right) ]
[ v_2’ = v_2 - \text{proj}_{u_1}(v_2) = \left(1,0,1\right) - \left(\frac{1}{2}, \frac{1}{2}, 0\right) = \left(\frac{1}{2}, -\frac{1}{2}, 1\right) ]
[ u_2 = \frac{v_2’}{|v_2’|} ]
[ = \frac{\left(\frac{1}{2}, -\frac{1}{2}, 1\right)}{\sqrt{\left(\frac{1}{2}\right)^2 + \left(-\frac{1}{2}\right)^2 + (1)^2}} ]
[ = \frac{\left(\frac{1}{2}, -\frac{1}{2}, 1\right)}{\sqrt{\frac{1}{4} + \frac{1}{4} + 1}} = \frac{\left(\frac{1}{2}, -\frac{1}{2}, 1\right)}{\sqrt{\frac{3}{2}}} ]
Step 3: Orthogonalize ( v_3 ) against ( u_1 ) and ( u_2 )
Follow the same procedure for ( v_3 ), projecting onto ( u_1 ) and ( u_2 ), then normalizing.
Summary
The Gram-Schmidt process transforms a set of linearly independent vectors into an orthonormal basis step by step, ensuring each vector is orthogonal to the previous ones and has unit length. This process is fundamental in applications such as:
- QR decomposition in numerical linear algebra.
- Constructing orthonormal bases for function spaces in Fourier analysis.
- Improving numerical stability in computational algorithms.
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