Final answer:
To find a basis for W₁, we need to find linearly independent vectors that span W₁. By reducing the given vectors to row echelon form and identifying the pivot columns, we can find a basis for W₁. To find an orthonormal basis for W₁, we need to normalize the vectors that are orthogonal to the basis vectors of W₁.
Step-by-step explanation:
To find a basis for a vector space, we need to find a set of vectors that are linearly independent and span the vector space. In this case, W is a subspace of R⁴ defined by W = Span{(1,2,-1,0), (0,-1,1,1), (1,0,1,2)}.
(a) To find a basis for W, we need to find the linearly independent vectors that span W. We can do this by reducing the given vectors to row echelon form and identifying the pivot columns. The reduced row echelon form of the given vectors is [(1, 0, 1, 2), (0, 1, -1, -1), (0, 0, 0, 0)]. The pivot columns are the first two columns. So, a basis for W is {(1, 2, -1, 0), (0, -1, 1, 1)}.
(b) To find a basis for W₁, we need to determine the vectors that are orthogonal to the basis vectors of W. We can use the dot product to check the orthogonality.
(c) To find an orthonormal basis for W, we need to normalize the basis vectors of W. Normalize each vector by dividing each component by its magnitude. Therefore, an orthonormal basis for W is {(1/√6, 2/√6, -1/√6, 0), (0, -1/√2, 1/√2, 1/√2)}.
(d) Similarly, to find an orthonormal basis for W₁, we need to normalize the vectors that are orthogonal to the basis vectors of W. Normalize these vectors by dividing each component by its magnitude. Therefore, an orthonormal basis for W₁ is {(-1/√2, 0, 1/√2, 0), (0, 1/√14, -1/√14, -1/√14)}.