By Eckhard M.S. Hitzer, University of Fukui, Prof. Ryosuke Nagaoka (University of the Air, Japan)

Now for this abstract concept of change of basis and the matrix which is used to express it, I will give you a concrete example. And in this example you see two aspects of this matrix. Once how it is expressing the change of basis and once how the same matrix is expressing a linear map, which has a different meaning.

So here you see the set of polynomials of at most second degree and the sign for it is the calligraphic f with index two ...

... Prof. Nagaoka ...

And we have the constants expressed with a, then the linear term b times x, and the quadratic term c times x squared. We can do the same examples also with higher orders, but just to limit the complexity, it doesn't change the content, but just to write down everything neatly we go up to second order. So this is expressed in the basis of constants, which are multiples of one, linear terms which are multiples of x, and a three, the quadratic terms, multiples of x squared. But we can equally express it with a different set of basic functions, for example a one prime taking one again, a two prime now as x minus 1, this is a two minus a one, and a three prime as x minus one squared, which is x squared minus two x plus one. So it is a three minus two a two plus a one. And we have the two basis and we can see how with a matrix we can express the second basis in terms of the first basis.

So and here you see these three equations, which I mentioned to you written down. A one prime, which is one (equals one) equals a one. A two prime is minus a one plus a two (it was minus one plus x). And a three prime is a one minus two a two plus a three. So this is one minus two x plus x squared.

And so these three equations we can express in matrix form, writing the basis vectors of the new basis, the basis vectors of the old basis and then the matrix, and we multiply this line always with one column and get one equation here. So this line with the second column gives this equation, and this row with the third column gives the third equation.

Now from this matrix, this matrix appears also in a different context. Now look at a linear map from the space of polynomials of second order into the space. But now we have a map where we have the coefficients a, b, c; and this expression is mapped into (the) another expression with the same coefficients, but they are multiplied with the new basis vectors. And so we see here this map and we can insert for the new basis vectors the explicit expressions here one, x minus one, and x minus one squared. And then we can multiply it out, and we can reexpress this term with the old basis: one, x and x squared. So coefficients are precisely a minus b plus c, b minus two c, and c. And so then the transformation from a, b, c to a minus b plus c, from a, b, c to b minus two c, and from a, b, c to c, is just expressed by this matrix.

And now I will show you another way how this matrix is used to express a linear map. And here we have the expression in the new basis: a prime (times) a one prime plus b prime (times) a two prime plus c prime (times) a three prime, and it shall be equal a a one plus b a two plus c a three. We take this expression and we insert the explicit forms of the basis. So the constant one, x minus one, x minus one squared. We multiply it out, and then we summarize it according (to the new) to the old basis. So one, x and x squared. So this is the old basis, a one, a two, and a three. And the coefficients, which we have here are precisely to be a, b and c. So a, b and c are a prime minus b prime plus c, b is b prime minus two c, and c is c prime. So these three euqations can conveniently (be) expressed again with the matrix. But as you see here, now the matrix is on the left side. And it shows you, if you change the basis from a one, a two, a three to a one prime, a two prime, a three prime, then the coefficients change in this way. With the help of this matrix you can make this calculation.

Yes thank you.

Soli Deo Gloria. Created by
Eckhard Hitzer (Fukui).

Last Modified 16 January 2004

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