- #1

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- 44

## Homework Statement

##\mathbb{H} = \{(a,b,c) : a - 3b + c = 0,~b - 2c = 0,~2b - c = 0 \}##

## Homework Equations

## The Attempt at a Solution

This definition of a subspace gives us the vector ##(3b - c,~2c,~2b) = b(3,0,2) + c(-1,2,0)##. This seems to suggest that a basis is {(3, 0, 2), (-1, 2 0)}, and that the subspace is 2-dimensional. However, if I take a different approach and solve the homogeneous system given by the subspace, we have that the only a, b and c that satisfy the system is (0, 0 , 0), which is to say that the subspace has no basis and is zero-dimensional. What am I doing wrong to get these two different answers?