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Unformatted text preview: Mods Linear Algebra Michaelmas Term 2011 The notes posted for this course on the web are not lecture notes. They are intended to complement the lectures and do not serve as a substitute for attending lectures. Lectures will try to focus on key ideas and important examples, with explanations of why the central results are important, and give outline proofs. The web notes will include details omitted from lectures and in particular complete proofs of the main theorems. Some examples will be treated in lectures which are not in the web notes, and vice versa. You are recommended to take some notes of your own in lectures. Page breaks are inserted between the numbered subsections of the web notes so that you can interleave your own handwritten notes. Web notes, in sections by topic, will be posted ahead of the relevant lectures. It will help to read through the web notes in advance to get a first impression of the content. KE, 1/10/11 1 1 Matrix algebra 1.1 Let m,n be positive integers. An m × n matrix is a rectangular array, with nm numbers, arranged in m rows and n columns. For example parenleftbigg 1 − 5 0 3 2 parenrightbigg is a 2 × 3 matrix. We allow the possibility of having just one column, or just one row, such as 3 / 2 9 17 , or ( − . 5 19 25 ) In general we write an m × n matrix as x 11 x 12 ... x 1 n x 21 x 22 ... x 2 n x 31 x 32 ... x 3 n . . . . . . ... . . . x m 1 x m 2 ... x mn We often abbreviate this and write X = [ x ij ] m × n or just [ x ij ] if it’s clear -or not important- what m and n are. Note that x ij appears in the i-th row and in the j-th column. The x ij are sometimes called ’matrix entries’. If A = [ a ij ] m × n and B = [ b ij ] p × q then A and B are defined to be equal , ie A = B if and only if (1) m = p and n = q ; and (2) a ij = b ij for all i,j . 1.1 Addition and scalar multiplication Definition 1.1 Suppose A and B are m × n matrices. Define the sum A + B to be the m × n matrix whose i,j entry is a ij + b ij . Note that A and B must have the same size. To find A + B , just add corresponding entries, and then A + B also has the same size. For example parenleftbigg 1 − 4 3 parenrightbigg + parenleftbigg − 1 2 3 4 parenrightbigg = parenleftbigg − 2 6 4 parenrightbigg If we write A + B then we assume implicitly that A and B have the same size. Definition 1.2 The m × n matrix with all entries equal to zero is called the zero matrix, and written as m × n or just as ....
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