Consider ℝ^3 with two orthonormal bases: the canonical basis e=(e1, e2, e3) and the basis f=(f1,

step 1 Chapter 2, exercise 48 of section 6. Suppose vb is equal to B sub 1, v .1, plus B sub 2, v .2, e

Consider ℝ^3 with two orthonormal bases: the canonical basis...

Consider $\mathbb{R}^{3}$ with two orthonormal bases: the canonical basis $e=\left(e_{1}, e_{2}, e_{3}\right)$ and the basis $f=\left(f_{1}, f_{2}, f_{3}\right)$, where $$ f_{1}=\frac{1}{\sqrt{3}}(1,1,1), f_{2}=\frac{1}{\sqrt{6}}(1,-2,1), f_{3}=\frac{1}{\sqrt{2}}(1,0,-1) $$ Find the matrix, $S$, of the change of basis transformation such that $$ [v]_{f}=S[v]_{e}, \text { for all } \mathrm{v} \in \mathbb{R}^{3} $$ where $[v]_{b}$ denotes the column vector of $v$ with respect to the basis $b$.

Consider $\mathbb{R}^{3}$ with two orthonormal bases: the canonical basis $e=\left(e_{1}, e_{2}, e_{3}\right)$ and the basis $f=\left(f_{1}, f_{2}, f_{3}\right)$, where
$$
f_{1}=\frac{1}{\sqrt{3}}(1,1,1), f_{2}=\frac{1}{\sqrt{6}}(1,-2,1), f_{3}=\frac{1}{\sqrt{2}}(1,0,-1)
$$
Find the matrix, $S$, of the change of basis transformation such that
$$
[v]_{f}=S[v]_{e}, \text { for all } \mathrm{v} \in \mathbb{R}^{3}
$$
where $[v]_{b}$ denotes the column vector of $v$ with respect to the basis $b$.

Key Concepts

Vector Spaces and Bases

In any vector space, a basis is a set of linearly independent vectors that spans the whole space. This allows each vector in the space to be uniquely expressed as a linear combination of these basis vectors. A change of basis involves expressing vectors with respect to a different set of basis vectors while retaining their overall representation in the vector space.

Change of Basis Transformation

This concept involves converting the coordinate representation of vectors from one basis to another using a transformation matrix. Specifically, if you have coordinates for a vector with respect to one basis, you can use the change of basis matrix to obtain its coordinates relative to a different basis by applying the matrix to the original coordinate column vector.

Orthonormal Basis

An orthonormal basis consists of vectors that are both orthogonal (perpendicular) to each other and of unit length. This property simplifies many computations because the inner products of distinct basis vectors vanish and the inner product of each vector with itself is one, which is advantageous in forming change of basis matrices, as the matrix becomes orthogonal (its inverse is its transpose).

Matrix Representation of Linear Transformations

A change of basis transformation can be represented by a matrix where each column is the coordinate representation of a new basis vector expressed in the old basis. This matrix directly translates the coordinates of any vector from the old representation to the new representation, encapsulating the linear relationship between the two different coordinate systems.

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