普通地质学3.5 Change of basis普通地质学.pdf

Change of basis Introduction In the previous section we talked about bases and dimension for a vector space. Many applied problems can be simplified by changing from one coordinate system to another. Changing coordinate systems in a vector space is Changing coordinate systems in a vector space is essentially the same as changing from one basis to another. In this section we discuss the problem of change of basis for vector spaces. We will show that this can be accomplished by multiplying a given coordinate vector by a nonsingular matrix and show an example. Outline 1. Definition of coordinate vectors 2. Change of basis in R2 3. Change of basis for a general vector space 4. Example Definition of coordinate vectors in R2 The standard basis for R2 is {e , e }. Any vector x in R2 1 2 can be expressed as a linear combination x x =e + x e . 1 1 2 2 The scalars x and x can be thought of as the 1 2 coordinates coordinates of x with respect to the standard basis. of x with respect to the standard basis. 2 Actually, for any basis {y, z} for R , a given vector x can be represented uniquely as x = αy + βz. The scalars α and β are the coordinates of x with respect to the basis {y, z}. We refer to the vector (α, β)T as the coordinate vector of x with respect to [y, z]. Change of basis in R2 Ex. Let 1 2 1       b1  ，b2  ，x  . 1 3 2       Find the coordinate vector y of x with respect to the basis {b , b }. 1 2 Solution. Suppose