2331_Notes_3o4_fill

2331_Notes_3o4_fill - Math 2331 Linear Algebra Section 3.4...

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Math 2331 Linear Algebra Section 3.4 The Complete Solution to Ax=b First, remember Ax in matrix times column vector is A LINEAR COMBINATION OF THE COLUMNS OF A If and 11 0 4 22 4 7 4431 A ⎛⎞ ⎜⎟ =− ⎝⎠ 1 2 3 4 x x x x x = then Ax = So Ax = b has a solution IF AND ONLY IF - b is The system Ax = b is called CONSISTENT if it has a solution (it may have more, but it needs to have at least one) The system Ax = b is called INCONSISTENT if it has no solution. This happens if b is NOT a linear combination of the columns of A. The system Ax = 0 is called the HOMOGENEOUS system corresponding to Ax = b. How to tell "which b's will work" Row reduce the augmented matrix [A|b} with a generic b. If there are zero rows on the bottom of the A side, the expression on the b side must be equal to zero for the system to be consistent. Example - find the conditions for the b vector such that the system is consistent - 1 2 3 12 25 38 b x b y b =
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11 22 33 44 102 4 028 4 014 2 000 0 x b x b x b x b ⎛⎞⎛⎞
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2331_Notes_3o4_fill - Math 2331 Linear Algebra Section 3.4...

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