# Does the dimension of the subspace increase by this operation?

lets say I have a finite dimensional(lets say 'k' is the dimension of the subspace) subspace over a vector space of dimension 'n' over binary field i.e {0,1}^n. Lets say I have added an extra parity bit to each vector from the subspace(dimension 'k') resulted from the XOR (exclusive OR) operation of all the bits of the vector. What happens to the dimension of the resulting new subspace. If 1) if 'k' is even 2) if 'k' is odd

• I think the dimension of the new subspace formed by adding an extra parity bit does not increase the dimension. But I am not sure. Could some one justify it or disprove it. – VKV Jun 22 '14 at 5:48

Let $V\subset \mathbb{F}_2^n$ denote your original space. Your new space $W\subset \mathbb{F}_2^{n+1}$ is the image of $V$ under the linear map $v \to v \oplus \left(\sum v_i\right)$. It's clearly injective, so it's an isomorphism onto $W$. Thus $\dim W = \dim V$.