SOLUTION: Find the basis for the following subspaces of R4 A. Vectors for which x1 = 2x4 B. Vectors for which x1 + x2 + x3 = 0 and x3 + x4 = 0

Question 1152518: Find the basis for the following subspaces of R4
A. Vectors for which x1 = 2x4
B. Vectors for which x1 + x2 + x3 = 0 and x3 + x4 = 0
Answer by rothauserc(4718) (Show Source): You can put this solution on YOUR website!
Call the subspace W. Then W consists of all the points (x1,x2,x3,x4) that are solutions to the equations
:
(A) x1 = 2x4, therefore
:
(2x4, x2, x3, x4) belongs to R^4 such that x2, x3, x4 belong to R
:
(x1, x2, x3, x4) = (2x4, x2, x3, x4)
:
(x1, x2, x3, x4) = x2 * (0, 1, 0, 0) +x3 * (0, 0, 1, 0) +x4 * (2, 0, 0, 1) and
:
the span = < (0, 1, 0, 0), (0, 0, 1, 0), (2, 0, 0, 1) >
:
In this case, the span is the basis
:
(B) x1 + x2 + x3 = 0 and x3 + x4 = 0
:
x1 = -x2 -x3 = -(x2 +x3) = x4 -x2
:
x3 = -x4
:
(x1, x2, x3, x4) = (x4-x2, x2, -x4, x4)
:
(x1, x2, x3, x4) = x2(-1, 1, 0, 0) +x4(1, 0, -1, 1)
:
basis is (-1, 1, 0, 0), (1, 0, -1, 1)
:

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