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l_notes31

# l_notes31 - Lecture XXXI Linear Equation Systems âŽ As we...

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Unformatted text preview: Lecture XXXI Linear Equation Systems âŽ¡ As we saw in the previous lecture, we can multiply m Ã— n matrices by column n-vectors. Consider the rows of an m Ã— n matrix A to be n-vectors: âŽ¤ âŽ¡ âŽ¤ âŽ¡ âŽ¤ r 1 c 1 r 1 C Â· âŽ¢ âŽ¢ âŽ¢ âŽ¢ âŽ¢ âŽ£ r 2 . . . âŽ¥ âŽ¥ âŽ¥ âŽ¥ âŽ¥ âŽ¦ , C = âŽ¢ âŽ¢ âŽ¢ âŽ¢ âŽ¢ âŽ£ c 2 . . . âŽ¥ âŽ¥ âŽ¥ âŽ¥ âŽ¥ âŽ¦ then AC = âŽ¢ âŽ¢ âŽ¢ âŽ¢ âŽ¢ âŽ£ r 2 âŽ¥ âŽ¥ âŽ¥ âŽ¥ âŽ¥ âŽ¦ C Â· . . A = . C r m c n r m Â· For example, if âŽ¤ âŽ¡ âŽ¤ âŽ¡ âŽ¤ âŽ¡ 7 1 2 âˆ’ 1 âŽ£ 27 22 âŽ¦ âŽ¢ âŽ¢ âŽ£ âŽ¥ âŽ¥ âŽ¦ , C = , then AC = âŽ£ âŽ¦ 9 A = 4 0 3 âˆ’ 2 We can use multiplication by a column vector to solve equation systems. An m Ã— n equation system has the form a 11 x 1 + a 12 x 2 + + a 1 n x n = d 1 Â·Â·Â· a 21 x 1 + a 22 x 2 + + a 2 n x n = d 2 Â·Â·Â· Â·Â·Â· Â·Â·Â· Â·Â·Â· Â·Â·Â· Â·Â·Â· a m 1 x 1 + a m 2 x 2 + + a mn x n = d m Â·Â·Â· This system has m equations and n unknowns: x 1 , x 2 , Â·Â·Â· , x n . An example of a 2 Ã— 3 system is 4 x 1 + 3 x 2 âˆ’ x 3 = 1 x 1 âˆ’ x...
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l_notes31 - Lecture XXXI Linear Equation Systems âŽ As we...

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