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It is given that X1, X2, ···, XM are M non-zero, orthogonal vectors. The dimension of the vector space spanned by the 2M vectors X1, X2, ···, XM, −X1, −X2, ···, −XM is

1. 2M
2. M + 1
3. M
4. dependent on the choice of X1, X2, ···, XM

Correct answer: M

Solution

The dimension of the vector space spanned by the vectors X1, X2, ···, XM and their negatives is determined by the number of linearly independent vectors. Since the original M vectors are orthogonal and non-zero, they are linearly independent, and the negatives do not introduce any new dimensions, resulting in a total dimension of M.

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