# Thread: Basis and dimension question

1. ## Basis and dimension question

Another basis questions I'm having trouble with:
Find basis for the following subsbace of R^5:
W1 = {(a1,a2,a3,a4,a5) which is an element of R^5: a1-a3-a4=0}
and
W2 = {(a1,a2,a3,a4,a5) which is an element of R^5: a2=a3=a4 and a1+a5=0}

What are the dimensions of W1 and W2?

2. Originally Posted by lllll
Another basis questions I'm having trouble with:
Find basis for the following subsbace of R^5:
W1 = {(a1,a2,a3,a4,a5) which is an element of R^5: a1-a3-a4=0}
and
W2 = {(a1,a2,a3,a4,a5) which is an element of R^5: a2=a3=a4 and a1+a5=0}

What are the dimensions of W1 and W2?

A typical element of W1 is (a3 + a4, a2, a3, a4, a5) (why?). One possible basis is therefore {(1, 0, 1, 0, 0), (1, 0, 0, 1, 0), (0, 1, 0, 0, 0), (0, 0, 0, 0, 1)}. Check that it spans and is linearly independent. The dimension of W1 is the number of vectors in its basis ......

A typical element of W2 is (a1, a2, a2, a2, -a1) (why?). One possible basis is therefore {(1, 0, 0, 0, -1), (0, 1, ,1 ,1, 0)}. Check that it spans and is linearly independent. The dimension of W2 is the number of vectors in its basis ...

3. Originally Posted by mr fantastic
A typical element of W1 is (a3 + a4, a2, a3, a4, a5) (why?). One possible basis is therefore {(1, 0, 1, 0, 0), (1, 0, 0, 1, 0), (0, 1, 0, 0, 0), (0, 0, 0, 0, 1)}. Check that it spans and is linearly independent. The dimension of W1 is the number of vectors in its basis ......

A typical element of W2 is (a1, a2, a2, a2, -a1) (why?). One possible basis is therefore {(1, 0, 0, 0, -1), (0, 1, ,1 ,1, 0)}. Check that it spans and is linearly independent. The dimension of W2 is the number of vectors in its basis ...
How did you actually get to {(1, 0, 1, 0, 0), (1, 0, 0, 1, 0), (0, 1, 0, 0, 0), (0, 0, 0, 0, 1)}?

4. Originally Posted by lllll
How did you actually get to {(1, 0, 1, 0, 0), (1, 0, 0, 1, 0), (0, 1, 0, 0, 0), (0, 0, 0, 0, 1)}?
Look at (a3 + a4, a2, a3, a4, a5) and think of a simple way of constructing it .......