By Garrity

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**Example text**

For example, if the above definition of a limit is to make sense, it must yield that lim x 2 = 4. x-t2 We will check this now. It must be emphasized that we would be foolish to show that x 2 approaches 4 as x approaches 2 by actually using the definition. We are again doing the common trick of using an example whose answer we already know to check the reasonableness of a new definition. Thus for any E > 0, we must find a 0 > 0 so that if 0 < Ix - 21 < 0, we will have Set o= min ( i, 1). As often happens, the initial work in finding the correct expression for 0 is hidden.

X R n to the real numbers. 5. THE DETERMINANT In order to be able to use this definition, we would have to prove that such a function on the space of matrices, satisfying conditions a through c, even exists and then that it is unique. Existence can be shown by checking that our first (inductive) definition for the determinant satisfies these conditions, though it is a painful calculation. The proof of uniqueness can be found in almost any linear algebra text. The third definition for the determinant is the most geometric but is also the most vague.

1 The eigenvalues for similar matrices are equal. Thus to see if two matrices are similar, one can compute to see if the eigenvalues are equal. If they are not, the matrices are not similar. Unfortunately in general, having equal eigenvalues does not force matrices to be similar. For example, the matrices and B = (~ ~) both have eigenvalues 1 and 2, but they are not similar. ) 18 CHAPTER 1. LINEAR ALGEBRA Since the characteristic polynomial P(t) does not change under a similarity transformation, the coefficients of P(t) will also not change under a similarity transformation.