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When n = 2 and when n = 3 all of this reduces to the vectors in the plane and in space that most of us learned in high school. The natural map from some Rn to an Rm is given by matrix multiplication. Write a vector x ERn as a column vector: x=CJ Similarly, we can write a vector in Rm as a column vector with m entries. ,tAy. 2. THE BASIC VECTOR SPACE R N · 3 In the next section we will use the linearity of matrix multiplication to motivate the definition for a linear transformation between vector spaces.

G) For all v E V, 1 . v = v. lThe real numbers can be replaced by the complex numbers and in fact by any field (which will be defined in Chapter Eleven on algebra). 3. VECTOR SPACES AND LINEAR TRANSFORMATIONS 5 As a matter of notation, and to agree with common usage, the elements of a vector space are called vectors and the elements of R (or whatever field is being used) scalars. Note that the space R n given in the last section certainly satisfies these conditions. The natural map between vector spaces is that of a linear transformation.

For each positive integer n, let b-a 6t = - n and a to, to + 6t, tl + 6t, tn-l For example, on the interval [0,2] with n to = 0 + 6t. = 4, we have 6t = 2"4 0 =~ and 4=2 On each interval [tk-l, tk], choose points lk and Uk such that for all points t on [tk-l, tk], we have and We make these choices in order to guarantee that the rectangle with base [tk-l, tk] and height f(h) is just under the curve y = f(x) and that the rectangle with base [tk-l, tk] and height f(Uk) is just outside the curve y = f(x).

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