The given vectors span a subspace W of R^3. Use the...

The given vectors span a subspace $W$ of $R^{3}$. Use the Gram-Schmidt orthogonalization process to construct an orthonormal basis for the subspace.
$$
\mathbf{u}_{1}=\langle 1,5,2\rangle, \mathbf{u}_{2}=\langle-2,1,1\rangle
$$

Key Concepts

Gram-Schmidt Orthogonalization

The Gram-Schmidt process is an algorithm used to convert a set of linearly independent vectors into an orthogonal or orthonormal set spanning the same subspace. It systematically subtracts projections of a vector onto the previously obtained orthogonal vectors and then normalizes the resulting vectors to obtain an orthonormal basis.

Projection

Projection is the process of decomposing a vector into a component that lies in the direction of another vector (or subspace) and a component that is orthogonal to that direction. In the context of the Gram-Schmidt process, projections are used to remove the components of vectors that are in the direction of the already constructed basis vectors, ensuring the orthogonality of the resulting set.

Subspace and Span

A subspace is a subset of a vector space that is also a vector space under the same addition and scalar multiplication. The span of a set of vectors is the collection of all possible linear combinations of those vectors, forming a subspace. This concept is fundamental in understanding how vectors can generate spaces in linear algebra.

Orthonormal Basis

An orthonormal basis is a set of vectors that are both orthogonal (each pair of distinct vectors has a dot product of zero) and normalized (each vector has unit length). This basis simplifies computations in many areas of linear algebra because it preserves lengths and angles, making it easier to decompose and analyze vectors.

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