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Unformatted text preview: Differential Equations & Linear Algebra Lecture 15 Jianli XIE Email: [email protected] Department of Mathematics, SJTU Fall 2010 Contents Vector space Contents Vector space Linear dependence and independence Contents Vector space Linear dependence and independence Basis and dimension Linear system of n equations in n unknowns Theorem If the coefficient matrix A of a system of n equations in n unknowns is nonsingular, then the system Ax = b has the unique solution x = A 1 b . Proof Since A is invertible, multiplying A 1 on both side of the equation, we obtain the result. Theorem Let A be a square matrix. If A is nonsingular, then the homogeneous system Ax = has only the trivial solution; If A is singular, then Ax = has nontrivial solutions. Definition of vector space Definition A vector space V is a set with two operations + and * that satisfy the following properties: a. If u and v are elements in V , then u + v is an element of V (we say V is closed under + ), and 1. u + v = v + u 2. u + ( v + w ) = ( u + v ) + w 3. There is an element in V such that u + = + u = u 4. For every u in V there is an element u with u + ( u ) = Definition of vector space b. If u is in V and c is a real number, then c u is in V (we say V is closed under * ), and 1. c * ( u + v ) = c * u + c * v 2. ( c + d ) * u = c * u + d * u 3. c * ( d * u ) = ( c * d ) * u 4. 1 * u = u Remark Obviously, the set of all n × 1 vectors (with real components) is a vector space, which is usually denoted by R n . There are other mathematical objects that also form vector spaces. Examples of vector spaces Example 1 Consider the set P 2 of all polynomials of degree less than or equal to 2 . We define + to be the usual addition of polynomials, and * be scalar...
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 Fall '10
 JianliXie
 Linear Algebra, Vector Space

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