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.

(a) (x)
(y) ∈R³
(z)
(b) [ (1) ]
[ (3) ]
[ (-1) ]
(c) (1) (1)
(0) (1)
(1) (1)
(2) (1)
(d) [0]

Answer 1

To find the orthogonal complement of a subspace represented by 2x - 3y + z = 0, one must solve for the null space of the matrix with these coefficients. Gaussian elimination is used to reduce the matrix, leading to the basis vectors for the orthogonal complement.

Step-by-step explanation:

To find the orthogonal complement of the given subspace in ℝ³, we use the equation 2x - 3y + z = 0. A vector v = (x, y, z) will be orthogonal to the subspace if it satisfies this equation when dotted with every vector in the subspace. This essentially means finding the null space of the matrix formed by the coefficients of the equation expressing the subspace.

By setting up the augmented matrix with the equation's coefficients (considering a zero vector on the right side of the equation), we can perform Gaussian elimination. This yields a matrix in reduced row echelon form. The null space of this matrix gives us the basis vectors for the orthogonal complement of the subspace. We assign free variables to x, y, and z and solve for them such that the dot product with any vector in the subspace is zero. After the calculations, the orthogonal complement is found, which is a set of all vectors that are orthogonal to every vector in the given subspace.