MAT223H1 Chapter Notes - Chapter 6.3: Linear Map, Linear Combination
Document Summary
In this section we look for a general procedure to switch from one basis of rn to another. The problem arises in many practical situations such as in physics problems, where the geometry of the situation will require a more suitable basis. Consider a linear transformation t : r2 r2 de ned by the matrix a, such that. Let us assume that we know the eigenvalues of a, with two linearly independent eigenvectors {u1, u2} and their corresponding eigenvalues { 1, 2}. Then {u1, u2} spans r2 and any vector v r2 can be writ- ten as linear combination of them: The complete coordinate expression can be denoted com- pactly using matrix algebra. Then, the coordinate change can be performed by (cid:3) un. U =(cid:2) u1 y = u y1 y2 yn. The matrix u is called a change of basis matrix which al- lows you to switch from basis to another.
