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A set of vectors in a vector space that are mutually orthogonal form a basis for the vector space. True False

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Answer:

False

Step-by-step explanation:

A set of vectors in a vector space that are mutually orthogonal does not necessarily form a basis for the vector space. In order for a set of vectors to form a basis, they must not only be mutually orthogonal but also linearly independent.
Two vectors are said to be orthogonal if their dot product is zero. Similarly, a set of vectors is mutually orthogonal if every pair of vectors in the set is orthogonal to each other.
On the other hand, for a set of vectors to be linearly independent, none of the vectors in the set can be written as a linear combination of the others.

In other words, no vector in the set can be expressed as a scalar multiple of another vector in the set.
Therefore, in order for a set of vectors to form a basis for a vector space, they must satisfy both the conditions of being mutually orthogonal and linearly independent.
To summarize:

- A set of mutually orthogonal vectors does not necessarily form a basis for a vector space.
- In addition to being mutually orthogonal, the vectors in a set must also be linearly independent in order to form a basis.

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