By Géza Schay
Construction at the author's previous edition at the topic (Introduction to Linear Algebra, Jones & Bartlett, 1996), this ebook bargains a refreshingly concise text suitable for the standard direction in linear algebra, presenting a carefully selected array of essential topics that may be completely lined in one semester. Although the exposition regularly falls in accordance with the material advised by the Linear Algebra Curriculum examine Group, it notably deviates in providing an early emphasis at the geometric foundations of linear algebra. this provides scholars a extra intuitive figuring out of the topic and permits an easier clutch of extra summary thoughts coated later within the path. the point of interest all through is rooted within the mathematical fundamentals, but the textual content also investigates a few attention-grabbing functions, together with a bit on computer graphics, a bankruptcy on numerical tools, and plenty of workouts and examples utilizing MATLAB. in the meantime, many visuals and difficulties (a whole ideas handbook is obtainable to teachers) are incorporated to augment and strengthen figuring out through the publication. short but special and rigorous, this work is a perfect selection for a one-semester path in linear algebra certain essentially at math or physics majors. It is a valuable tool for any professor who teaches the topic.
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Extra info for A Concise Introduction to Linear Algebra
Then p and q are orthogonal4 to each other if and only if p · q = 0. 2. He was led to these names by geometrical considerations. “Orthogonal” is synonymous with “perpendicular” but is, for some reason, preferred in linear algebra. 18 1. Analytic Geometry of Euclidean Spaces P p-q p O q Q Fig. 15. A right triangle in terms of vectors Proof. Assume ﬁrst p ⊥ q. 15). 26) from which p · q = 0 follows at once. 24. 24 implies that the OP Q triangle is a right triangle with p ⊥ q, provided p = 0 and q = 0.
Am1 am2 · · · amn where the a11 , a12 , . . ) are arbitrary real numbers. They are called the entries of the matrix A, with aij denoting the entry at the intersection of the ith row and jth column. Thus in the double subscript ij the order is important. Also, the matrix A is often denoted by [aij ] or (aij ). Two matrices are said to be equal if they have the same shape, that is, the same numbers of rows and columns, and their corresponding entries are equal. A matrix consisting of a single row is called a row vector, and that of a single column, a column vector , and, if we want to emphasize the size n, a row n-vector or a column n-vector.
42) 22 1. 5. 6. 43) with equality holding if and only if p is parallel to q. Proof. 35, and if p = 0 or q = 0 or both, then the result is trivially true with 0 = 0. ) Thus all that remains to show is that equality holds for nonzero vectors if and only if p is parallel to q. 5, |cos θ| = 1, and so θ = 0 or θ = π, and in both cases p is parallel to q. Conversely, if p and q are nonzero and p is parallel to q, then p = aq, for some scalar a, and so |p · q| = |a| |q · q| = |a| |q|2 and |p||q| = |aq||q| = |a| |q|2 , that is, the two sides in Cauchy’s inequality are equal.
A Concise Introduction to Linear Algebra by Géza Schay