Question

It is given that $X_1,X_2,...X_M$ are M non-zero, orthogonal vectors. The dimension of the vector space spanned by the 2M vectors $X_1,X_2,...X_M,-X_1,-X_2,...-X_M$ is

• A. 2M
• B. M + 1
• C. M
• D. dependent on the choice of $X_1,X_2,...X_M$

Answer 1

The correct option is C M

Since $x_1,x_2,...x_m$ are orthogonal they span a vector space of dimension M.

Since $-x_1,-x_2,-x_3,...,-x_m$ are linearly dependent on $x_1,x_2,...,x_m$ so the set ( $x_1,x_2,x_3,...,x_m$) will span a vector space of dimension M only