Question

Find a basis for the orthogonal complement of the following. A few may be easy to do by inspection. You may proceed as outlined in the text: Solve the equations that arise from orthonality to a basis. In this method, we turn the basis of the space into the rows of a matrix and find the null space of the matrix.

[x]
(a) 2x - 3y +z = 0
[z]

Answer 1

To find the basis for the orthogonal complement of the space defined by the equation 2x - 3y + z = 0, we look for vectors that are orthogonal to (2, -3, 1), resulting in the basis vectors (1, 0, -2) and (0, 1, 3).

Step-by-step explanation:

To find a basis for the orthogonal complement of the space defined by the equation 2x - 3y + z = 0 in ℝ³, we consider vectors in the space that are orthogonal to every vector that satisfies this equation. This space is a plane in three-dimensional space, and its orthogonal complement will be a line through the origin, which is normal to the plane.

We use the dot product to determine orthogonality. If we have a vector (x,y,z) that is orthogonal to the plane, its dot product with any vector in the plane will be zero. The coefficients of the given equation can be seen as a vector (2,-3,1) which is normal to the plane. Hence, we are looking for vectors that have a dot product of zero with this normal vector.

Let's say our vector in the orthogonal complement is (a,b,c). The dot product with the plane's normal vector gives us 2a - 3b + c = 0. We can set a and b equal to arbitrary parameters and solve for c to express it in terms of a and b, yielding the basis vector or vectors for the orthogonal complement. By setting a = 1, b = 0, we get c = -2, and by setting a = 0, b = 1, we get c = 3. Therefore, the basis for the orthogonal complement is the set containing the vectors (1,0,-2) and (0,1,3).