The subspace has dimension 3.
Intuitively, you can think that you're removing one degree of freedom from [tex] \mathbb{R}^4 [/tex]. In fact, this space is generated by the following span:
[tex] \lambda_1 (1,0,0,0) + \lambda_2 (0,1,0,0) + \lambda_3 (0,0,1,0) + \lambda_4 (0,0,0,1) [/tex]
Since the first and third coordinate must be the same, we have [tex] \lambda_1 = \lambda_3 [/tex]. So, the span becomes
[tex] \lambda_1 (1,0,0,0) + \lambda_2 (0,1,0,0) + \lambda_1 (0,0,1,0) + \lambda_4 (0,0,0,1) [/tex]
which is the same as
[tex] \lambda_1 (1,0,1,0) + \lambda_2 (0,1,0,0) + \lambda_4 (0,0,0,1) [/tex]
And so you can see that the space is generated by three vectors.
