{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

LinAlgNotes

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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 .
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}