Show that {𝐮1, 𝐮2} or {𝐮1, 𝐮2, 𝐮3} is an orthogonal basis for ℝ^2 or ℝ^3, respectively. Then exp

step 1 Okay, this question wants us to show that this is an orthogonal basis for R3, and then we'll use

Show that {𝐮1, 𝐮2} or {𝐮1, 𝐮2, 𝐮3} is an orthogonal basis...

Show that $\left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\}$ or $\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\}$ is an orthogonal basis for $\mathbb{R}^{2}$ or $\mathbb{R}^{3},$ respectively. Then express $\mathbf{x}$ as a linear combination of the $\mathbf{u}^{\prime}$s. $\mathbf{u}_{1}=\left[\begin{array}{l}{1} \\ {0} \\ {1}\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{r}{-1} \\ {4} \\ {1}\end{array}\right], \mathbf{u}_{3}=\left[\begin{array}{r}{2} \\ {1} \\ {-2}\end{array}\right],$ and $\mathbf{x}=\left[\begin{array}{r}{8} \\ {-4} \\ {-3}\end{array}\right]$

Show that $\left\{\mathbf{u}_{1}, \mathbf{u}_{2}\right\}$ or $\left\{\mathbf{u}_{1}, \mathbf{u}_{2}, \mathbf{u}_{3}\right\}$ is an orthogonal basis for $\mathbb{R}^{2}$ or $\mathbb{R}^{3},$ respectively. Then express $\mathbf{x}$ as a linear combination of the $\mathbf{u}^{\prime}$s.
$\mathbf{u}_{1}=\left[\begin{array}{l}{1} \\ {0} \\ {1}\end{array}\right], \mathbf{u}_{2}=\left[\begin{array}{r}{-1} \\ {4} \\ {1}\end{array}\right], \mathbf{u}_{3}=\left[\begin{array}{r}{2} \\ {1} \\ {-2}\end{array}\right],$ and $\mathbf{x}=\left[\begin{array}{r}{8} \\ {-4} \\ {-3}\end{array}\right]$

Key Concepts

Orthogonal Basis

An orthogonal basis is a set of vectors in a vector space that are mutually perpendicular—meaning that the inner product (or dot product in Euclidean spaces) between any pair of distinct vectors is zero. When a basis is orthogonal, every vector in the space can be expressed uniquely as a linear combination of these basis vectors, and the coefficients in this linear combination can be computed simply using the inner product divided by the square norm of each basis vector.

Inner Product

The inner product is a key operation in vector spaces that provides a way to define length and angle. In Euclidean spaces, it is commonly defined as the dot product. It is instrumental in testing for orthogonality (two vectors are orthogonal if their inner product is zero) and in finding the coefficients of a vector when expressing it as a linear combination of an orthogonal basis through projection.

Linear Combination

A linear combination involves expressing one vector as the sum of scalar multiples of other vectors. In the context of bases, any vector in the space can be uniquely represented as a linear combination of the basis vectors. The process is simplified when using an orthogonal basis because the coefficients can be calculated independently, often using projection formulas derived from the inner product.

Projection

Projection refers to the method of breaking down a vector into components along the directions determined by a set of basis vectors. When the basis is orthogonal, the projection of a vector onto each basis vector provides the coefficient for that direction in the linear combination representation of the vector. This is achieved by computing the inner product of the vector with the basis vector and dividing by the square of the norm of the basis vector.

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