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LinAlgNotes - Notes on Linear Algebra These notes are...

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Notes on Linear Algebra January 18, 2010 These notes are intended to give a brief overview of some key terms and ideas from linear algebra, without all the technical details. For full details see any linear algebra textbook. 1 Vector spaces, inner products A vector space V is a set of objects that can be added to one another or multiplied by any scalar belonging a given number field F , to yield another object in V . Usually F = R or F = C (the complex numbers), although in principle F could be another field, such as the rational numbers. For present purposes, let’s take the scalar field F to be the real numbers R . Here are some examples of vector spaces. 1. V = R 3 . This serves as a model for ordinary space. We denote elements x, y R 3 by a column vectors x = x 1 x 2 x 3 , y = y 1 y 2 y 3 , x + y = x 1 + y 1 x 2 + y 2 x 3 + y 3 . Given any scalar α R , the product αx R 3 is a vector. 2. A more general family of examples is V = R n , where n 1 is any fixed natural number. If n = 1 we have the real numbers themselves as an example of a vector space, and if n = 3 we have the previous example. 1
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For general n , we still represent elements x, y R n as column vectors x = x 1 x 2 . . . x n , y = y 1 y 2 . . . y n , x + y = x 1 + y 1 x 2 + y 2 . . . x n + y n . Scalar multiplication works as in the previous example: for any α R and x R n , αx = αx 1 αx 2 .
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