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.