Question

Consider the vector v = (3, 2, 1) in R³. Let W be the subspace of R³ consisting of all vectors of the form (a, b, a b). Decompose v into the sum of a vector that lies in W and a vector orthogonal to W.

Answer 1

To decompose the vector v = (3, 2, 1) into the sum of a vector that lies in W and a vector orthogonal to W, we first need to find a basis for the subspace W.

Step-by-step explanation:

To decompose the vector v = (3, 2, 1) into the sum of a vector that lies in W and a vector orthogonal to W, we first need to find a basis for the subspace W. A vector (a, b, ab) lies in W if and only if it is a linear combination of the basis vectors. Let's choose (1, 0, 0) and (0, 1, 0) as the basis vectors for W. Therefore, any vector in W can be written as:

(a, b, ab) = a(1, 0, 0) + b(0, 1, 0)

We can rewrite the vector v in terms of this basis:

(3, 2, 1) = 3(1, 0, 0) + 2(0, 1, 0) + 1(0, 1, 0)

The vector (3, 2, 1) can be decomposed into the sum of a vector that lies in W and a vector orthogonal to W as follows:

(3, 2, 1) = 3(1, 0, 0) + 2(0, 1, 0) + 0(0, 1, 0) + 1(0, 0, 1)

The first three terms form a vector in W, and the last term is orthogonal to W.