Use the Gram-Schmidt process to construct an orthonormal...

Use the Gram-Schmidt process to construct an orthonormal basis for the following subspaces of $\mathbb{R}^3$ : (a) the plane spanned by $(0,2,1)^T,(1,-2,-1)^T ;$ (b) the plane defined by the equation $2 x-y+3 z=0 ;(c)$ the set of all vectors orthogonal to $(1,-1,-2)^T$.

Key Concepts

Gram-Schmidt Process

A method used to convert a set of linearly independent vectors into an orthogonal or orthonormal set while keeping the span unchanged. This process involves iteratively subtracting projections of the current vector onto the previously obtained orthogonal vectors, ensuring that each new vector is orthogonal to all the ones before it.

Orthonormal Basis

A basis for a subspace in which all the vectors are mutually orthogonal and each vector has unit length. An orthonormal basis simplifies computations in many areas of linear algebra, including solving systems of equations and performing orthogonal projections.

Subspace

A subset of a vector space that is itself a vector space, meaning it is closed under vector addition and scalar multiplication. Examples include lines and planes in ?^3, which can be defined either by a spanning set of vectors or by linear equations.

Orthogonal Projection

A technique used to decompose a vector into a component that lies in the direction of another vector (or subspace) and a component that is orthogonal to it. This concept is crucial in the Gram-Schmidt process for removing the part of a vector that is in the direction of preceding orthogonalized vectors.

Normalization

The process of converting a nonzero vector into a unit vector by dividing it by its norm. This step is essential after obtaining an orthogonal set in order to form an orthonormal basis, ensuring all basis vectors have length one.

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