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Unformatted text preview: Mathematics V2010y Linear Algebra Spring 2007 PROVING STATEMENTS IN LINEAR ALGEBRA Linear algebra is different from calculus: you cannot understand it properly without some simple proofs. Knowing statements of formulas isnt enough. You also have to know their contexts, and how to deduce one from another. Thats what these notes are intended to help with. But proof is as much an art as a science, so these are merely guidelines. What is a proof? Just a convincing argument to explain why a mathematical statement is true. However, long experience has led us to develop very clear standards for what is convincing and what isnt. We will try to explain the steps that most frequently arise in a typical proof. But let us be clear: a proof does not have to be one of those mechanical constructions from high school with statements on the left and reasons on the right. Rather, it should be an argument (preferably in complete sentences) that is intelligible and convincing to a human being. Lets start by considering what kinds of basic objects we are dealing with. 1 What are our basic objects of study? There are two answers. Scalars, vectors, and matrices. First, there are real numbers, also known as scalars . Then there are m n matrices . There are also vectors in R m , but we regard them as m 1 matrices. You might say that scalars should also be regarded as 1 1 matrices. We might sometimes want to bend the rules this way, but its usually better not to. For example, if is a scalar and A is a 2 2 matrix, then the product A is defined, but you cant multiply a 1 1 matrix by a 2 2 matrix. In any equation, the same type of object should appear on both sides. Its an error to equate a scalar to a vector, for example. Sets. The other basic objects that we consider are sets . This is what many proofs are about. A set is just any collection of elements , which can be any kind of objects: scalars, vectors, matrices, or even apples and oranges. Its an error to equate a set with an element. For example, Span is a set, not a single vector, so its an error to write Span ( u , v ) = u + v . An example of a set is R , the set of all real numbers....
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