Discussion
What is the dimension of the subspace spanned by 0111, 1010, 0011 and 1110 using generator matrix ?
Answer: D
The dimension is 3.
Working over GF(2) (addition is XOR), check for linear dependence. The fourth vector is the sum of the other three:
\[ \begin{aligned} 0111 + 1010 &= 1101 \\ 1101 + 0011 &= 1110 \end{aligned} \]
So 1110 adds nothing new. The first three are independent: no single one is zero, no two are equal, and no two add to the third (0111 + 1010 = 1101, 0111 + 0011 = 0100, 1010 + 0011 = 1001). Three independent vectors span a subspace of dimension 3, with 8 codewords; dimensions 0, 1 and 2 are too small.
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