Let TA: R^3 →R^3 be multiplication by the orthogonal matrix...

Let $T_{A}: R^{3} \rightarrow R^{3}$ be multiplication by the orthogonal matrix in Exercise $5 .$ Find $T_{A}(\mathbf{x})$ for the vector $\mathbf{x}=(-2,3,5),$ and confirm that $\left\|T_{A}(\mathbf{x})\right\|=\|\mathbf{x}\|$ relative to the Euclidean inner product on $R^{3}$.

Key Concepts

Euclidean Inner Product

The Euclidean inner product is a function that assigns a real number to a pair of vectors in Euclidean space, capturing the geometric notion of the dot product. It is fundamental in defining notions of length (or norm) and angles between vectors. This inner product is the basis for the standard Euclidean geometry in R^n and is critical in studying the behavior of vectors under linear transformations.

Norm Preservation

Norm preservation refers to the property of certain linear transformations—specifically, those arising from orthogonal matrices—to maintain the length of vectors. When a transformation preserves the Euclidean norm, it ensures that distances and angles remain unchanged. This is a central concept when dealing with rotations and reflections, as these do not distort the geometry of the space.

Orthogonal Matrices

Orthogonal matrices are square matrices whose columns (and rows) are orthonormal vectors. This property implies that the inverse of an orthogonal matrix is equal to its transpose and that such matrices preserve the Euclidean inner product between vectors. They play a key role in transformations that maintain lengths and angles, making them essential in various applications of linear algebra, computer graphics, and physics.

Linear Transformations

A linear transformation is a mapping between vector spaces that preserves the operations of vector addition and scalar multiplication. In the context of matrix multiplication, each matrix defines a linear transformation that operates on vectors. Understanding how these transformations act on vectors allows for the exploration of geometric properties such as rotation, reflection, and scaling.

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