By Géza Schay
Building at the author's earlier version at the topic (Introduction toLinear Algebra, Jones & Bartlett, 1996), this booklet deals a refreshingly concise textual content appropriate for the standard path in linear algebra, offering a gently chosen array of crucial themes that may be completely lined in one semester. even though the exposition ordinarily falls in keeping with the fabric steered through the Linear Algebra Curriculum research staff, it significantly deviates in offering an early emphasis at the geometric foundations of linear algebra. this offers scholars a extra intuitive figuring out of the topic and permits a neater clutch of extra summary strategies coated later within the path.
The concentration all through is rooted within the mathematical basics, however the textual content additionally investigates a couple of fascinating functions, together with a piece on special effects, a bankruptcy on numerical tools, and plenty of routines and examples utilizing MATLAB. in the meantime, many visuals and difficulties (a entire options handbook is out there to teachers) are incorporated to augment and strengthen realizing in the course of the publication.
Brief but designated and rigorous, this paintings is a perfect selection for a one-semester path in linear algebra particular basically at math or physics majors. it's a necessary software for any professor who teaches the subject.
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Additional resources for A Concise Introduction to Linear Algebra
Consider the system 2x1 + 3x2 − 2x3 + 4x4 = 2 −6x1 − 9x2 + 7x3 − 8x4 = −3 4x1 + 6x2 − x3 + 20x4 = 13. We solve this system as follows: ⎤ ⎡ ⎡ 2 r1 ← r1 2 2 3 −2 4 ⎣ −6 − 9 7 − 8 −3 ⎦ r2 ← r2 + 3r1 ⎣ 0 4 6 −1 20 13 r3 ← r3 − 2r1 0 ⎤ ⎡ r1 ← r1 2 2 3 −2 4 ⎣0 3⎦. 31) Since the pivots are in columns 1 and 3, the basic variables are x1 and x3 and the free variables x2 and x4 . Thus we use two parameters and set x2 = s and x4 = t. Then the second row of the last matrix leads to x3 = 3−4t and the ﬁrst row to 2x1 +3s−2(3−4t)+4t = 2, that is, to 2x1 = 8−3s−12t.
32) ⎣ −4 ⎦ ⎣ 0 ⎦ ⎣ x3 ⎦ ⎣ 3 ⎦ 1 0 0 x4 which is also a parametric vector equation of a plane in R4 . Exercises In the ﬁrst four exercises, ﬁnd all solutions of the systems by Gaussian elimination. 1. 2. 2x1 + 2x2 − 3x3 = 0 x1 + 5x2 + 2x3 = 0 6x3 = 0 −4x1 + 52 2. 3. 4. 2x1 + 2x2 − 3x3 = 0 In the next nine exercises use Gaussian elimination to ﬁnd all solutions of the systems given by their augmented matrices. 5. 6. 7. 8. 9. 10. 11. 12. 13.
A Plane Through Two Lines). 3. 3. If we take P0 = (4, 3, 9), say, then S will be described by the equation p = (4, 3, 9) + s(2, −3, 7) + t(−1, 4, 2). 77 and eliminate s and t: x = 4 + 2s − t, y = 3 − 3s + 4t, z = 9 + 7s + 2t. 3 Lines and Planes 33 Bring the constant terms to the left, multiply the ﬁrst of these equations in turn by 4 and 2, and add the results to the second and third equations respectively, to get 4(x − 4) + (y − 3) = 5s and 2(x − 4) + (z − 9) = 11s. 80) Now multiply the ﬁrst of these equations by 11 and the second one by (−5), and add the results.