SL2 (R) by Serge Lang

By Serge Lang

SL2(R) offers the coed an advent to the countless dimensional illustration thought of semisimple Lie teams by means of targeting one instance - SL2(R). This box is of curiosity not just for its personal sake, yet for its connections with different parts resembling quantity idea, as introduced out, for instance, within the paintings of Langlands. The quick improvement of illustration concept over the last forty years has made it more and more tricky for a pupil to go into the sphere. This publication makes the speculation obtainable to a large viewers, its basically must haves being an information of actual research, and a few differential equations.

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Of course, planes can be described in different ways. 14 Chapter 1 Vectors and Matrices EXAMPLE 9 We wish to find a parametric equation of the plane that contains the points P = (1, 2, 1) −→ and Q = (2, 4, 0) and is parallel to the vector (1, 1, 3). We take x0 = (1, 2, 1), u = P Q = (1, 2, −1), and v = (1, 1, 3), so the plane consists of all points of the form x = (1, 2, 1) + s(1, 2, −1) + t (1, 1, 3), s, t ∈ R. Finally, note that three noncollinear points P , Q, R ∈ R3 determine a plane. To get a −→ −→ −→ parametric equation of this plane, we simply take x0 = OP , u = P Q, and v = P R.

If c1 , . . , ck ∈ R, the vector v = c1 v1 + c2 v2 + · · · + ck vk is called a linear combination of v1 , . . , vk . 15 12 Chapter 1 Vectors and Matrices Definition. Let v1 , . . , vk ∈ Rn . The set of all linear combinations of v1 , . . , vk is called their span, denoted Span (v1 , . . , vk ). That is, Span (v1 , . . , vk ) = {v ∈ Rn : v = c1 v1 + c2 v2 + · · · + ck vk for some scalars c1 , . . , ck }. In terms of our new language, then, the span of two nonparallel vectors u, v ∈ Rn is a plane through the origin.

16 Chapter 1 Vectors and Matrices 6. Find a parametric equation of each of the following lines: a. 3x1 + 4x2 = 6 ∗ b. the line with slope 1/3 that passes through A = (−1, 2) c. the line with slope 2/5 that passes through A = (3, 1) d. the line through A = (−2, 1) parallel to x = (1, 4) + t (3, 5) e. the line through A = (−2, 1) perpendicular to x = (1, 4) + t (3, 5) ∗ f. the line through A = (1, 2, 1) and B = (2, 1, 0) g. the line through A = (1, −2, 1) and B = (2, 1, −1) ∗ h. the line through (1, 1, 0, −1) parallel to x = (2 + t, 1 − 2t, 3t, 4 − t) 7.

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