Question

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

Construct the matrix A that transforms each vj into uj to give Av1 = u1, . . . , Avn = un.

Answer 1

To construct the matrix A transforming orthonormal basis vectors
`\( \mathbf{v}_j \)` to
`\( \mathbf{u}_j \)`, use
`\( A = [\mathbf{u}_1, \ldots, \mathbf{u}_n] [\mathbf{v}_1, \ldots, \mathbf{v}_n]^T \)`, where
`\( [\mathbf{u}_1, \ldots, \mathbf{u}_n] \)` and
`\( [\mathbf{v}_1, \ldots, \mathbf{v}_n] \)` are matrices of basis vectors.

To construct the matrix A that transforms each vector
`\( \mathbf{v}_j \)` into
`\( \mathbf{u}_j \)`, we can use the following approach:

`\[ A = [\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_n] [\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n]^(-1) \]`

Here,
`\( [\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_n] \) is the matrix whose columns are the vectors \( \mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_n \), and \( [\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n] \) is the matrix whose columns are the vectors \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n \)`.

Given that
`\( \{\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_n\} \)` and
`\( \{\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n\} \)` are orthonormal bases, the inverse of
`\( [\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n] \)` is simply its transpose. Therefore:

`\[ A = [\mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_n] [\mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_n]^T \]`

This gives the matrix A that transforms each vector
`\( \mathbf{v}_j \)` into
`\( \mathbf{u}_j \)`.