Decide if each statement is True or False.
1. If the set $\left\{\vec{u}_{1}, \vec{u}_{2}, \vec{u}_{3}\right\}$ is orthogonal, then it must be linearly independent. choose your answer...
2. If the set $\left\{\vec{u}_{1}, \vec{u}_{2}, \vec{u}_{3}\right\}$ is linearly independent, then it must be orthogonal. choose your answer...
3. If the set $\left\{\vec{u}_{1}, \vec{u}_{2}, \vec{u}_{3}\right\}$ is orthonormal, then it must be linearly independent. choose your answer...
4. The standard basis $\left\{\vec{e}_{1}, \ldots, \vec{e}_{n}\right\}$ of $\mathbb{R}^{n}$ is an orthonormal basis of $\mathbb{R}^{n}$. choose your answer...
5. If the set $\left\{\vec{u}_{1}, \vec{u}_{2}, \vec{u}_{3}\right\}$ is orthonormal, then $V=\operatorname{Span}\left\{\vec{u}_{1}, \vec{u}_{2}, \vec{u}_{3}\right\}$ must have $\operatorname{dim}(V)=3$. choose your answer...
Answer
Final Answer: The statement "If the set \(\left\{\vec{u}_{1}, \vec{u}_{2}, \vec{u}_{3}\right\} is orthogonal, then it must be linearly independent" is \(\boxed{True}\).
Step 1 ：The first statement is asking if an orthogonal set of vectors is necessarily linearly independent. Orthogonal vectors are vectors that are at right angles to each other. In other words, the dot product of any two vectors in the set is zero. Linearly independent vectors are vectors that cannot be written as a linear combination of the other vectors in the set. If a set of vectors is orthogonal, then it must be linearly independent. This is because if a vector could be written as a linear combination of the other vectors, then it would not be at right angles to them. Therefore, the first statement is True.
Step 2 ：Final Answer: The statement "If the set \(\left\{\vec{u}_{1}, \vec{u}_{2}, \vec{u}_{3}\right\} is orthogonal, then it must be linearly independent" is \(\boxed{True}\).
