MATH-M 303 Lecture Notes - Lecture 3: Scalar Multiplication, Row And Column Vectors, Zero Element
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M303 section 1. 3 notes- vectors and vector equations. Vectors allow new interpretations of linear systems and their solutions. A matrix with 1 column (ie. (cid:883)) known as column vector: set of all such vectors denoted by , vectors (cid:2206)=[(cid:1876)(cid:2869)(cid:1876)(cid:2870)(cid:1709)(cid:1876)] and (cid:2207)=[(cid:1877)(cid:2869)(cid:1877)(cid:2870)(cid:1709)(cid:1877)] are equal iff their corresponding entries are equal (ie. [(cid:887)(cid:883)]=[(cid:887)(cid:883)] but. [(cid:887)(cid:883)] [(cid:883)(cid:887)]: sometimes written as regular vectors for space: (cid:2206)=(cid:4666)(cid:1876)(cid:2869),(cid:1876)(cid:2870), ,(cid:1876)(cid:4667, vectors in (cid:2870) and (cid:2871) can be interpreted as points in plane/space, resp, zero vector of - (cid:2777)=(cid:4666)(cid:882),(cid:882), ,(cid:882)(cid:4667, sum of vectors (cid:2206)=(cid:4666)(cid:1876)(cid:2869),(cid:1876)(cid:2870), ,(cid:1876)(cid:4667) and (cid:2207)=(cid:4666)(cid:1877)(cid:2869),(cid:1877)(cid:2870), ,(cid:1877)(cid:4667): (cid:2206)+(cid:2207)=(cid:4666)(cid:1876)(cid:2869),(cid:1876)(cid:2870), ,(cid:1876)(cid:4667)+(cid:4666)(cid:1877)(cid:2869),(cid:1877)(cid:2870), ,(cid:1877)(cid:4667) =(cid:4666)(cid:1876)(cid:2869)+(cid:1877)(cid:2869),(cid:1876)(cid:2870)+(cid:1877)(cid:2870), ,(cid:1876)+(cid:1877)(cid:4667: scalar multiple (cid:1855)(cid:2206) for scalar (cid:1855) : (cid:1855)(cid:2206)=(cid:1855)(cid:4666)(cid:1876)(cid:2869),(cid:1876)(cid:2870), ,(cid:1876)(cid:4667) =(cid:4666)(cid:1855)(cid:1876)(cid:2869),(cid:1855)(cid:1876)(cid:2870), ,(cid:1855)(cid:1876)(cid:4667: algebraic properties (for (cid:2203),(cid:2204),(cid:2205) and (cid:1855),(cid:1856) ) =(cid:4666)(cid:882),(cid:882),(cid:882)(cid:4667) applies to any vectors if all (cid:1855)=(cid:882: order of scalar multiplications matters, order of adding does not, any/all vectors and/or scalars can be zero multiplications and taking the sum. Can be solved like normal augmented matrix: ex.
