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notes1-3 - vertex of the parallelogram whose other vertices...

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SECTION 1.3 VECTOR EQUATIONS Until further notice, a vector will be a matrix with exactly one column. Vectors with two rows can be identified with points in the plane; vectors with three rows can be identified with points in three-dimensional space. The collection of all n -row vectors is denoted by R n . We can add and subtract vectors of the same size as usual, entry-by-entry, and we can multiply vectors by numbers. We write vectors either using boldface type or, by hand, with arrows above them to distinguish them from just numbers. Every size has a zero vector, written 0 , all of whose entries are 0. Be careful, there is a big difference between 0 and 0. Vectors have very nice algebraic properties, as long as we restrict ourselves to addition, subtraction, and scalar multiplication. These properties are in a box on page 32. So, we can write equations that involve vectors. We can go back and forth between vector equations and systems of equations! If vectors u and v are represented as points, then the sum u + v corresponds to the fourth
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Unformatted text preview: vertex of the parallelogram whose other vertices are , u , and v . EXAMPLE. Write a system of equations that is equivalent to the vector equation x 1 -2 3 5 + x 2 1 2 = 4 1 3 . A linear combination of a list of vectors is a vector formed by adding multiples of the vectors in the list. The multiples used are also called weights . The set of all linear combinations of a list of vectors is called the subset spanned or generated by the vectors in the list. We denote this set by Span { v 1 , v 2 ,..., v p } . EXAMPLE. List three vectors in Span 1 3-2 , 3-1 4 . EXAMPLE. Is b = 1 3 1 a linear combination of the vectors formed from the columns of the matrix A = 1 3-2 5 2-3-9 6 ? Describe the subset spanned by these vectors. HOMEWORK: SECTION 1.3...
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