Question

Find the dimension of the subspace of all vectors in set of real numbers r superscript 4ℝ4 whose firstfirst and thirdthird entries are equal.

Answer 1

The subspace has dimension 3.

Intuitively, you can think that you're removing one degree of freedom from $\mathbb{R}^4$. In fact, this space is generated by the following span:

$\lambda_1 (1,0,0,0) + \lambda_2 (0,1,0,0) + \lambda_3 (0,0,1,0) + \lambda_4 (0,0,0,1)$

Since the first and third coordinate must be the same, we have $\lambda_1 = \lambda_3$. So, the span becomes

$\lambda_1 (1,0,0,0) + \lambda_2 (0,1,0,0) + \lambda_1 (0,0,1,0) + \lambda_4 (0,0,0,1)$

which is the same as

$\lambda_1 (1,0,1,0) + \lambda_2 (0,1,0,0) + \lambda_4 (0,0,0,1)$

And so you can see that the space is generated by three vectors.