Final answer:
To find the basis of the orthogonal complement W^⊥, which consists of vectors that are orthogonal to all vectors in W, we deduced the conditions for the orthogonal vector [a, b, c] to have a zero dot product with any vector [x, y, x + y] in W. Solving these conditions, we determined that a basis for W^⊥ is the set {[-1, -1, 1]}.
Step-by-step explanation:
The student is asking to find a basis for the orthogonal complement of the set W containing all vectors of the form [x, y, x + y] with x and y being real numbers. To find a vector that is orthogonal to every vector in W, we need to find a vector that has a dot product of zero with any vector in W. Let's denote our orthogonal vector as [a, b, c]. For it to be orthogonal to [x, y, x + y], the dot product must be (a*x) + (b*y) + (c*(x + y)) = 0 for all values of x and y.
To satisfy this condition for all x and y, we can deduce that:
- a + c = 0 (so the x terms cancel out)
- b + c = 0 (so the y terms cancel out)
From these equations, it's clear that a = -c and b = -c. Choosing c to be 1, we get the vector [-1, -1, 1] which is the basis for W⊥ as it is orthogonal to all vectors in W. Therefore, the set {[-1, -1, 1]} is a basis of W⊥.