By Kenneth Kuttler

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There is a much easier way and it is this which is presented here. It is also important to note that not all matrices have inverses. 22 Let A = 1 1 1 1 . Does A have an inverse? One might think A would have an inverse because it does not equal zero. However, 1 1 1 1 −1 1 = 0 0 and if A−1 existed, this could not happen because you could write 0 0 0 0 = A−1 −1 1 = A−1 A −1 1 = A−1 A −1 1 =I −1 1 = = , a contradiction. Thus the answer is that A does not have an inverse. 23 Let A = 1 1 1 2 2 −1 . Show −1 1 is the inverse of A.

If so, prove it. 5. Let f, g be continuous functions and define 1 f (t) g (t)dt f ·g ≡ 0 show this satisfies the axioms of a dot product if you think of continuous functions in the place of a vector in Fn . What does the Cauchy Schwarz inequality say in this case? 6. Show that if f is a real valued continuous function, 2 b f (t) dt a ≤ (b − a) b 1/2 2 f (t) dt. 1 Matrices You have now solved systems of equations by writing them in terms of an augmented matrix and then doing row operations on this augmented matrix.

Proof: This follows immediately from the above theorem. 24 depends only on these vectors. This theorem shows that any linear transformation defined on Fn can always be considered as a matrix. Therefore, the terms “linear transformation” and “matrix” are often used interchangeably. For example, to say a matrix is one to one, means the linear transformation determined by the matrix is one to one. 6 Find the linear transformation, L : R2 → R2 which has the property that 2 1 Le1 = and Le2 = . 4 1 3 .

### An Introduction To Linear Algebra by Kenneth Kuttler

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