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lect1102Matrices

# lect1102Matrices - MATH 17100 28446 Matrices Recall the...

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MATH 17100 28446 11/02/2009 Matrices Recall the vectors we defined as n -tuples, namely , ordered collections of numbers, for which we devised and defined operations of addition, multiplication by a scalar, dot-multiplication and cross-multiplication, using them in spatial geometry to describe lines and curves, planes and surfaces. We now continue to extend and expand the use of vectors in solving systems of linear equations in algebra . We first adopt some new forms of expression for the vector. Given the vector with n components, 12 ( , , , ) n v v v v , we may express the same quantity as [ ] n v v v and call it a row vector . We may also express it as 1 2 n v v v    and call it a column vector . Notice that the commas delineating the components in our previous notation have been dropped, the components placed within square parentheses are separated by blank spaces when the components are listed horizontally in a row , and the vector is called a row vector, or they are placed on separate lines along a vertical column , and the vector is called a column vector. With this distinction, equality and addition of vectors require that they be compatible , namely that they have the same number of components and being of the same type, i.e. row- or column-. Consider the row vector with n components. Suppose each scalar component were replaced by a column vector with m components: | | | | | | n v v v , , 1, 2, , , k kn v being the th k column vector along the row, 11 1 21 22 2 n n m m m n v v v v v v v v v where, expressing explicitly the components of each column vector k v , we find that ik v is the th i component of the k column vector, or, judging from the rectangular array of quantities, we can say that v is the quantity found at the intersection of the

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th i row and th k column. The same rectangular array would result if we had started with a column vector with m components, and replaced each component by a row vector with n components: 1 2 m 
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lect1102Matrices - MATH 17100 28446 Matrices Recall the...

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