Suppose u1, …, un and v1, …, vn are orthonormal bases for 𝐑^n. Construct the matrix A that trans

step 1 For exercise 6, we have matrix A equal to U sub 1 v sub 1 transposed, U sub 1 v sub 2 transpose.

Suppose u1, …, un and v1, …, vn are orthonormal bases for...

Suppose $u_{1}, \ldots, u_{n}$ and $v_{1}, \ldots, v_{n}$ are orthonormal bases for $\mathbf{R}^{n}$. Construct the matrix $A$ that transforms each $v_{j}$ into $u_{j}$ to give $A v_{1}=u_{1}, \ldots, A v_{n}=u_{n}$.

Key Concepts

Orthogonal Matrices

Orthogonal matrices are square matrices with orthonormal columns (and rows), which preserve length and angles under transformation. In the context of mapping between orthonormal bases, the resulting transformation matrix is orthogonal, meaning its inverse is equal to its transpose (A?¹ = A?), a property that simplifies analysis and inversion.

Change of Basis and Matrix Construction

When two orthonormal bases are given, one for the domain and one for the codomain, the transformation matrix that maps one basis to the other can be constructed by forming matrices whose columns are the basis vectors. If U has the target basis vectors as its columns and V has the original basis vectors, then the transformation matrix is given by A = U V?. This construction uses the fact that V? extracts the coordinates relative to the original basis and U reassembles the vector in the coordinates of the new basis.

Matrix Representation of Linear Transformations

Any linear transformation on ?? can be represented by an n×n matrix. The transformation's action is fully determined by its effect on a basis. Knowing how a transformation maps a set of basis vectors allows one to construct its matrix representation by aligning the transformed vectors appropriately, often exploiting the structure provided by orthonormality.

Orthonormal Bases

An orthonormal basis in ?? is a set of n vectors that are all unit vectors and mutually orthogonal. This property simplifies many calculations in linear algebra because the inner product between different basis vectors is zero and the norm of each is one, which leads to simpler representations of linear transformations.

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