Change of Basis - Change of Coordinates of A Vector - Two Dimensions

Two Dimensions

This means that given a matrix M whose columns are the vectors of the new basis of the space (new basis matrix), the new coordinates for a column vector v are given by the matrix product M-1.v. For this reason, it is said that normal vectors are contravariant objects.

Any finite set of vectors can be represented by a matrix in which its columns are the coordinates of the given vectors. As an example in dimension 2, a pair of vectors obtained by rotating the standard basis counterclockwise for 45 degrees. The matrix whose columns are the coordinates of these vectors is

M=\begin{pmatrix} 1/\sqrt{2} & -1/\sqrt{2} \\ 1/\sqrt{2} & 1/\sqrt{2} \end{pmatrix}

If we want to change any vector of the space to this new basis, we only need to left-multiply its components by the inverse of this matrix.

Read more about this topic:  Change Of Basis, Change of Coordinates of A Vector

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