Answer:
To find a basis for the orthogonal complement of the subspace spanned by v1, v2, v3, and v4, we need to find a basis for the subspace of vectors in R^5 that are orthogonal to all four of these vectors.
We can use the Gram-Schmidt process to find an orthogonal basis for the subspace spanned by v1, v2, v3, and v4. Let's call this basis {u1, u2, u3, u4}. Then, any vector in the orthogonal complement of this subspace will be orthogonal to u1, u2, u3, and u4.
Using the Gram-Schmidt process, we can find an orthogonal basis for the subspace spanned by v1, v2, v3, and v4 as follows:
u1 = v1 = (1, 4, 5, 6, 9)
u2 = v2 - proj_u1(v2) = v2 - ((v2 · u1) / (u1 · u1))u1 = (3, -2, 1, 4, -1) - ((3 + 16 + 5 + 24 + 81) / (1 + 16 + 25 + 36 + 81))(1, 4, 5, 6, 9) = (-2, -10, -4, -10, -28)
u3 = v3 - proj_u1(v3) - proj_u2(v3) = v3 - ((v3 · u1) / (u1 · u1))u1 - ((v3 · u2) / (u2 · u2))u2 = (-1, 0, -1, -2, -1) - ((-1 + 0 - 5 - 12 - 9) / (1 + 16 + 25 + 36 + 81))(1, 4, 5, 6, 9) - ((-2 + 0 + 4 + 8 + 28) / (4 + 100 + 16 + 100 + 784))(-2, -10, -4, -10, -28) = (-11/5, -2/5, -6/5, 6/5, -9/5)
u4 = v4 - proj_u1(v4) - proj_u2(v4) - proj_u3(v4) = v4 - ((v4 · u1) / (u1 · u1))u1 - ((v4 · u2) / (u2 · u2))u2 - ((v4 · u3) / (u3 · u3))u3 = (2, 3, 5, 7, 8) - ((2 + 12 + 25 + 42 + 72) / (1 + 16 + 25 + 36 + 81))(1, 4, 5, 6, 9) - ((-6 - 12 - 4 - 40 + 28) / (4 + 100 + 16 + 100 + 784))(-2, -10, -4, -10, -28) - ((-22/5 - 6/5 + 6/5 - 42/5 + 18/5) / (121/25 + 4/25 + 36/25 + 36/25 + 81/25))(-11/5, -2/5, -6/5, 6/5, -9/
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Explanation: