In Excrcises 1-6, the given set is a basis for a subspace W...

In Excrcises $1-6,$ the given set is a basis for a subspace $W .$ Use the Gram-Schmidt process to produce an orthogonal basis for $W$ .
$$
\left[\begin{array}{r}{3} \\ {0} \\ {-1}\end{array}\right],\left[\begin{array}{r}{8} \\ {5} \\ {-6}\end{array}\right]
$$

Key Concepts

Subspace and Basis

A subspace is a subset of a vector space that is closed under addition and scalar multiplication. A basis of a subspace is a set of linearly independent vectors that spans the entire subspace. Transforming a basis into an orthogonal basis, as achieved through the Gram-Schmidt process, allows for more convenient and simplified computations.

Projection

Projection in linear algebra involves finding the component of one vector along the direction of another vector. This concept is crucial in the Gram-Schmidt process, where the projection of a vector onto the space spanned by previously processed vectors is subtracted from the vector to make it orthogonal to the earlier ones.

Orthogonal Basis

An orthogonal basis is a set of vectors within a subspace where each pair of distinct vectors is perpendicular, meaning their inner product is zero. This property simplifies many computations in linear algebra, such as solving systems of equations and performing projections, because calculations involving orthogonal vectors tend to decouple.

Gram-Schmidt Process

The Gram-Schmidt process is an algorithm used to convert a set of linearly independent vectors into an orthogonal (or orthonormal) set that spans the same subspace. By sequentially subtracting the projections of a vector onto the previously obtained orthogonal vectors, the process removes components in the direction of those vectors, ensuring mutual orthogonality.

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