More on linear algebra

In the previous slide we stated that a unitary transformation keeps the orthogonality, as discussed in exercise 1 below. To see this consider first a basis of vectors \( \mathbf{v}_i \), $$ \mathbf{v}_i = \begin{bmatrix} v_{i1} \\ \dots \\ \dots \\v_{in} \end{bmatrix} $$ We assume that the basis is orthogonal, that is $$ \mathbf{v}_j^T\mathbf{v}_i = \delta_{ij}. $$ An orthogonal or unitary transformation $$ \mathbf{w}_i=\mathbf{U}\mathbf{v}_i, $$ preserves the dot product and orthogonality since $$ \mathbf{w}_j^T\mathbf{w}_i=(\mathbf{U}\mathbf{v}_j)^T\mathbf{U}\mathbf{v}_i=\mathbf{v}_j^T\mathbf{U}^T\mathbf{U}\mathbf{v}_i= \mathbf{v}_j^T\mathbf{v}_i = \delta_{ij}. $$