Find the basis and dimension of the subspace. W = {(x,y,z): x,y,z ER and 2x+y+3z = 0) of a real vector space R
Question
Find the basis and dimension of the subspace.
W = {(x,y,z): x,y,z ∈ ℝ and 2x+y+3z = 0} of a real vector space ℝ.
Solution
The given subspace W is defined by the equation 2x + y + 3z = 0. We can rewrite this equation in terms of one variable, say y, to get y = -2x - 3z.
This means that any vector in W can be written in the form (x, -2x - 3z, z) = x(1, -2, 0) + z(0, -3, 1).
So, the vectors (1, -2, 0) and (0, -3, 1) form a basis for the subspace W.
The dimension of the subspace W is the number of vectors in its basis, so the dimension of W is 2.
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