Gram-Schmidt Process Calculator

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Example

Created on 2024-06-20Asked by William Garcia (Solvelet student)

Use the Gram-Schmidt process to orthogonalize the set of vectors

\mathbf{v}_1 = \begin{pmatrix} 1 \\ 1 \end{pmatrix}

and

\mathbf{v}_2 = \begin{pmatrix} 1 \\ 0 \end{pmatrix}

Solution

To use the Gram-Schmidt process to orthogonalize the set of vectors

\mathbf{v}_1 = \begin{pmatrix} 1 \ 1 \end{pmatrix}

and

\mathbf{v}_2 = \begin{pmatrix} 1 \ 0 \end{pmatrix}

: 1. Let

\mathbf{u}_1 = \mathbf{v}_1 = \begin{pmatrix} 1 \ 1 \end{pmatrix}

. 2. Compute the projection of

\mathbf{v}_2

onto

\mathbf{u}_1

\text{proj}_{\mathbf{u}_1} \mathbf{v}_2 = \frac{\mathbf{v}_2 \cdot \mathbf{u}_1}{\mathbf{u}_1 \cdot \mathbf{u}_1} \mathbf{u}_1

= \frac{\begin{pmatrix} 1 \ 0 \end{pmatrix} \cdot \begin{pmatrix} 1 \ 1 \end{pmatrix}}{\begin{pmatrix} 1 \ 1 \end{pmatrix} \cdot \begin{pmatrix} 1 \ 1 \end{pmatrix}} \begin{pmatrix} 1 \ 1 \end{pmatrix}

= \frac{1 \cdot 1 + 0 \cdot 1}{1^2 + 1^2} \begin{pmatrix} 1 \ 1 \end{pmatrix}

= \frac{1}{2} \begin{pmatrix} 1 \ 1 \end{pmatrix}

= \begin{pmatrix} \frac{1}{2} \ \frac{1}{2} \end{pmatrix}

3. Subtract the projection from

\mathbf{v}_2

to find

\mathbf{u}_2

\mathbf{u}_2 = \mathbf{v}_2 - \text{proj}_{\mathbf{u}_1} \mathbf{v}_2

= \begin{pmatrix} 1 \ 0 \end{pmatrix} - \begin{pmatrix} \frac{1}{2} \ \frac{1}{2} \end{pmatrix}

= \begin{pmatrix} \frac{1}{2} \ -\frac{1}{2} \end{pmatrix}

The orthogonal set is:

\mathbf{u}_1 = \begin{pmatrix} 1 \ 1 \end{pmatrix}, \quad \mathbf{u}_2 = \begin{pmatrix} \frac{1}{2} \ -\frac{1}{2} \end{pmatrix}

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Elijah Lee on Solvelet

1. Apply the Gram-Schmidt process to the set of vectors

\{(1, 0, -1), (1, 1, 0), (0, 1, 1)\}

\mathbb{R}^3

,2. Orthogonalize the vectors

(1, 1, 0)

and

(0, 1, 1)

using the Gram-Schmidt process.,

DefinitionThe Gram-Schmidt process is a process for orthonormalizing a set of vectors, generating the orthogonal mass that spans the supplements. This is a repeated subtraction of the projection of the space’s existing basis vectors and orthogonalizing the vectors, based on the set’s product memory space. However, in this case, the Gram-Schmidt process is as opposed to the inner product memory space, the Euclidean storage image Rn as the most common SPACE with the specialized image is a point in this space...

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