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Advanced Math/column space and linear combinations


Any help would be greatly appreciated. I a having difficulty understanding this problem.

Take an m by n matrix A. Let vector b be a vector in R^m. Assume that

vector x = <a_1,...,a_n>

is a solution for the matrix equation A(vector x) = vector b. Show that vector b is in the column space of A by writing vector b as a linear combination of the columns of A.

For A an mxn matrix and x an nx1 vector, the product Ax should be an mx1 vector, by the rules of matrix multiplication. For A, the mxn form means m rows and n columns, so the n columns of A represent vectors with m components. Therefore we can write

[ a11  a12 ... a1n   ] {x1]   [b1]
[ a21  a22 ... a2n   ] [x2]= [b2]
[  .          ]   .         .

[ am1 am2 ... amn ]          [bm]


[ a11]        [a12]       ... [a1n]        = [b1]
[ a21][x1]  [a22][x2] ... [a2n][xn  ]= [b2]
[  .    ]        [  .  ]          [  .   ]       = [  .  ]
[ am1]          .          [amn       = [bm].

where the components of the vector x serve as coefficients for the column vectors of A so give a linear combination for b.

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randy patton


college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography


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J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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