Orthogonal Basis and Coordinates in R3

Show that the given vectors form an orthogonal basis for $\mathbb{R}^3$. Then, express the given vector $\mathbf{w}$ as a linear combination of these basis vectors. Give the coordinates of vector $\mathbf{w}$ with respect to the orthogonal basis.

To demonstrate that a set of vectors forms an orthogonal basis for a space like $R^3$, you need to verify that each pair of vectors from the set is orthogonal to each other. This involves taking the dot product of each pair and ensuring that it equals zero. Once you've established orthogonality, the set of vectors forms a basis if they span $R^3$, which generally means there are three independent vectors in the set. This orthogonal basis provides the advantage of making calculations, such as projections and coordinates, more straightforward.

Once you have confirmed that the vectors are indeed an orthogonal basis, the next task involves expressing another vector, w, as a linear combination of these basis vectors. This requires calculating the coordinates or coefficients of w with respect to the orthogonal basis. In essence, you are transforming w into a new coordinate system defined by the orthogonal basis. The calculation simplifies because, for orthogonal bases, the coefficients can be calculated using simple projections of w onto each of the basis vectors.

Understanding orthogonal bases is crucial in linear algebra as it simplifies many operations, particularly those involving transformations and projections. These concepts also have applications in computer graphics, physics, and any field that requires efficient computation in multi-dimensional space.