the vectorS form an orthog Express x as a linear combination of the u's. u, + U3 (Use integers or fractions for any numbers in the equation.) U2
Unitary Method
The word “unitary” comes from the word “unit”, which means a single and complete entity. In this method, we find the value of a unit product from the given number of products, and then we solve for the other number of products.
Speed, Time, and Distance
Imagine you and 3 of your friends are planning to go to the playground at 6 in the evening. Your house is one mile away from the playground and one of your friends named Jim must start at 5 pm to reach the playground by walk. The other two friends are 3 miles away.
Profit and Loss
The amount earned or lost on the sale of one or more items is referred to as the profit or loss on that item.
Units and Measurements
Measurements and comparisons are the foundation of science and engineering. We, therefore, need rules that tell us how things are measured and compared. For these measurements and comparisons, we perform certain experiments, and we will need the experiments to set up the devices.
Help needed on the last part only
![### Orthogonal Basis for \( R^3 \)
**Problem Statement:**
Given vectors \( \{u_1, u_2, u_3\} \), show that they form an orthogonal basis for \( R^3 \). Then express \( x \) as a linear combination of the \( u \)'s.
\[ u_1 = \begin{pmatrix} 2 \\ -2 \\ 0 \end{pmatrix}, \quad u_2 = \begin{pmatrix} 3 \\ -1 \\ 1 \end{pmatrix}, \quad u_3 = \begin{pmatrix} 1 \\ 6 \\ -1 \end{pmatrix}, \quad \text{and} \quad x = \begin{pmatrix} 5 \\ -2 \\ -1 \end{pmatrix} \]
**Necessary Criteria:**
Which of the following criteria are necessary for a set of vectors to be an orthogonal basis for a subspace \( W \) of \( R^n \)? Select all that apply.
- \( \checkmark \) A. The vectors must form an orthogonal set.
- ☐ B. The vectors must all have a length of 1.
- ☐ C. The distance between any pair of distinct vectors must be constant.
- \( \checkmark \) D. The vectors must span \( W \).
**Theorems for Orthogonality:**
Which theorem could help prove one of these criteria from another?
- **A.** If \( S = \{u_1, u_2, u_3\} \) and the distance between any pair of distinct vectors is constant, then the vectors are evenly spaced and hence form an orthogonal set.
- **B.** If \( S = \{u_1, u_2, u_3\} \) and each \( u_i \) has length 1, then \( S \) is an orthogonal set and hence is a basis for the subspace spanned by \( S \).
- **C.** If \( S = \{u_1, u_2, u_3\} \) is a basis in \( R^n \), then the members of \( S \) span \( R^n \) and hence form an orthogonal set.
- \( \checkmark \) **D.** If \(](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F05b5783f-5507-4009-950e-cf9b7485abe1%2F52ccaef9-de85-4381-ac80-21b39662fa15%2Fm7twlba_processed.jpeg&w=3840&q=75)
Step by step
Solved in 2 steps with 2 images
