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December 30

matrices

in the multiplication of two matrix. why we multiply rows elements to column elements to getting answer — Preceding unsigned comment added by Dev follower of maths (talk • contribs) 05:00, 30 December 2010 (UTC)[reply]

The best way to think of matrices is that they're a way of coding linear transformations between vector spaces. Then matrix multiplication corresponds to taking the composition of two linear transformations. If you don't have that piece of the puzzle, the whole thing looks kind of unmotivated. --Trovatore (talk) 05:39, 30 December 2010 (UTC)[reply]

You can try this out for yourself using reflections and rotations as in our article on Matrix transformations. Try combining rotations of 90 degrees with reflections in a plane to make the arithmetic easy. You will see that multiplying the matrices gives the correct matrix for the combined transformation, and that the order of multiplication is often important. Dbfirs 20:35, 30 December 2010 (UTC)[reply]

I'm guessing that the OP wants to know about the use of matrices in applied mathematics. For example, take a look at pages 1, 2, 3, … of this book. — Fly by Night (talk) 22:22, 30 December 2010 (UTC)[reply]

Linear transformations are the right way to think about them, whether your aims are applied or not. Otherwise it just looks arbitrary. --Trovatore (talk) 22:43, 30 December 2010 (UTC)[reply]

Did you take a look at that link? I can't really put the applied uses in terms of linear algebra. Matrices with entries m_i,j where m_i,j is the cost of product i from warehouse j. Then you multiply it by matrices that involve tax and stuff like that. In accountancy they use matrices to work out profit margin over repeated monthly cycles, including tax and labour, etc. I don't think we can frame that in terms of vector spaces and linear algebra in any meaningful way. In those applications it's just a new kind of multiplication. It's a means to an end. — Fly by Night (talk) 03:52, 31 December 2010 (UTC)[reply]

If you can't put that in terms of linear algebra, you're not trying hard enough. --Trovatore (talk) 03:53, 31 December 2010 (UTC)[reply]

Maybe you could help me by explaining how... — Fly by Night (talk) 18:39, 31 December 2010 (UTC)[reply]

Excellent question! Think of it like this:

{\begin{aligned}u&=3x-5y\\v&=7x+8y\\\\\\p&=2u+51v\\q&=-3u+2v\\\\{\text{Therefore}}\\p&=({\text{?}})x+({\text{?}})y\\q&=({\text{?}})x+({\text{?}})y.\end{aligned}}

That's matrix multiplication. When you figure out how to fill in the blanks, then you have the answer to your question. Michael Hardy (talk) 01:12, 31 December 2010 (UTC) </math>[reply]