Thread: Find bases for the following subspaces of F^5

1. Find bases for the following subspaces of F^5

Find bases for the following subspaces of F^5:

W1 = {(a1, a2, a3, a4, a5) E F^5 : a1 - a3 - a4 = 0}

and

W2 = {(a1, a2, a3, a4, a5) E F^5: a2 = a3 = a4 and a1 + a5 = 0}

Well, I understand a basis is the maximum amount of vectors in a set that are linearly independent, or the smallest amount of L.I vectors that span a space. What is throwing me off is the constraints a1 - a3 - a4 = 0 and a2 = a3 = a4 and a1 + a5 = 0

2. Originally Posted by zodiacbrave
Find bases for the following subspaces of F^5: W1 = {(a1, a2, a3, a4, a5) E F^5 : a1 - a3 - a4 = 0}

Using $a_1=a_3+a_4$ express $(a_1,a_2,a_3,a_4,a_5)\in W_1$ as

$a_2(\;\ldots\+a_3(\;\ldots\+a_4(\;\ldots\+a_5(\;\ldots\" alt="a_2(\;\ldots\+a_3(\;\ldots\+a_4(\;\ldots\+a_5(\;\ldots\" />

Those four vectors span $W_1$ and it is easy to prove they are linearly independent.

W2 = {(a1, a2, a3, a4, a5) E F^5: a2 = a3 = a4 and a1 + a5 = 0}

Same method solving previously the system.

Fernando Revilla