Inner product spaces - Orthogonal complement, Finite - Dimensional subspace

(c) Inner product spaces - Orthogonal complement, finite - dimensional subspace.

Problem 1.

Solution :

We can also have

Therefore, we have the solution set which is the required  which can be given as :

Problem 2.

Let S₀ = {x₀}, where x₀ is a nonzero vector in R³. Describe  geometrically. Now suppose that S = {x₁, x₂} is a linearly independent subset of R³. Describe  geometrically.

Solution :

Let S₀ = {x₀}, where x₀ is a nonzero vector in R³.

Assume x₀ as a direction.

Therefore, by definition,  consists of two linearly independent vectors.

A plane, orthogonal to the direction defined by the vector x₀, is constructed by span of these two vectors.

Let S = {x₁,x₂}. Assume that one plane is constructed by spanning of the vectors contained in S.

Therefore, by definition,  consists of a single element. A linear span of this element is a line.

Therefore,  represents a line orthogonal to the plane which can be generated by the linear span of the elements of S.

Problem 3.

Let W₁ and W₂ be subspaces of a finite-dimensional inner product space.

Prove that 

Solution :

Let W₁ and W₂ be subspaces of a finite-dimensional inner product space.

To prove that

Proof :

Proceeding in a similar way, we can find that x is an element in  and therefore,