## 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 $$\mathbf{w}_j^T\mathbf{w}_i=(\mathbf{U}\mathbf{v}_j)^T\mathbf{U}\mathbf{v}_i=\mathbf{v}_j^T\mathbf{U}^T\mathbf{U}\mathbf{v}_i= \mathbf{v}_j^T\mathbf{v}_i = \delta_{ij}.$$