### Orthogonal Basis Using Gram-Schmidt Process**Objective:**To find the orthogonal basis for a subspace $ W $ using the Gram-Schmidt process.**Given:**The following set of vectors form a basis for a subspace $ W $:\[\mathbf{x_1} =\begin{pmatrix}1 \\-1 \\0 \\1\end{pmatrix}, \quad\mathbf{x_2} =\begin{pmatrix}3 \\-2 \\-7 \\1\end{pmatrix}\]**Step-by-Step Process:**1. **Starting with $\mathbf{x_1}$:**Set $\mathbf{u_1} = \mathbf{x_1}$.2. **Making $\mathbf{x_2}$ orthogonal to $\mathbf{u_1}$:**Compute the projection of $\mathbf{x_2}$ onto $\mathbf{u_1}$:\[\text{proj}_{\mathbf{u_1}}\mathbf{x_2} = \frac{\mathbf{x_2} \cdot \mathbf{u_1}}{\mathbf{u_1} \cdot \mathbf{u_1}} \mathbf{u_1}\]Subtract this projection from $\mathbf{x_2}$ to get $\mathbf{u_2}$:\[\mathbf{u_2} = \mathbf{x_2} - \text{proj}_{\mathbf{u_1}}\mathbf{x_2}\]**Outcome:**The orthogonal basis produced using the Gram-Schmidt process for $ W $ is as follows:\[\mathbf{u_1} = \begin{pmatrix}1 \\-1 \\0 \\1\end{pmatrix}, \quad\mathbf{u_2} = \text{(Calculated as per the steps above)}\]For the actual values of $\mathbf{u_2}$, you would proceed with the calculation steps explicitly to find the specific orthogonal vectors.**Note:** Remember to use a comma to separate the vectors when listing the orthogonal basis vectors.---This example illustrates the application of the Gram-Schmidt process to transform a basis into an orthogonal basis, enhancing understanding for students studying linear algebra concepts on an

The Gram-Schmidt Orthogonalization Process: Let V be a vector space with an inner product or dot…

The set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W. Assume the vectors are in the order x, and x,. 1 3 1 2 - 7 1 1 The orthogonal basis produced using the Gram-Schmidt process for W is { }. (Use a comma to separate vectors as needed.)

The set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W. Assume the vectors are in the order x, and x,. 1 3 1 2 - 7 1 1 The orthogonal basis produced using the Gram-Schmidt process for W is { }. (Use a comma to separate vectors as needed.)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

### Orthogonal Basis Using Gram-Schmidt Process

**Objective:**
To find the orthogonal basis for a subspace $ W $ using the Gram-Schmidt process.

**Given:**
The following set of vectors form a basis for a subspace $ W $:

\[
\mathbf{x_1} =
\begin{pmatrix}
1 \\
-1 \\
0 \\
1
\end{pmatrix}
, \quad
\mathbf{x_2} =
\begin{pmatrix}
3 \\
-2 \\
-7 \\
1
\end{pmatrix}
\]

**Step-by-Step Process:**

1. **Starting with $\mathbf{x_1}$:**

Set $\mathbf{u_1} = \mathbf{x_1}$.

2. **Making $\mathbf{x_2}$ orthogonal to $\mathbf{u_1}$:**

Compute the projection of $\mathbf{x_2}$ onto $\mathbf{u_1}$:

\[
\text{proj}_{\mathbf{u_1}}\mathbf{x_2} = \frac{\mathbf{x_2} \cdot \mathbf{u_1}}{\mathbf{u_1} \cdot \mathbf{u_1}} \mathbf{u_1}
\]

Subtract this projection from $\mathbf{x_2}$ to get $\mathbf{u_2}$:

\[
\mathbf{u_2} = \mathbf{x_2} - \text{proj}_{\mathbf{u_1}}\mathbf{x_2}
\]

**Outcome:**
The orthogonal basis produced using the Gram-Schmidt process for $ W $ is as follows:

\[
\mathbf{u_1} =
\begin{pmatrix}
1 \\
-1 \\
0 \\
1
\end{pmatrix}
, \quad
\mathbf{u_2} = \text{(Calculated as per the steps above)}
\]

For the actual values of $\mathbf{u_2}$, you would proceed with the calculation steps explicitly to find the specific orthogonal vectors.

**Note:** Remember to use a comma to separate the vectors when listing the orthogonal basis vectors.

---

This example illustrates the application of the Gram-Schmidt process to transform a basis into an orthogonal basis, enhancing understanding for students studying linear algebra concepts on an

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