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Unformatted text preview: g row operations to find the general solution X = X_{0} + x_{i} V_{i} + …+ x_{j} V_{j} as a sum of a particular solution X_{0} (always the zero vector in the case B = 0 of a homogeneous system) plus an arbitrary linear combination of independent solutions V_{k}, one for each free parameter x_{k}. But if we want to test a set A_{1}, .. ,A_{n} of column vectors to see if a vector B is in their SPAN, we try to solve the vector equation x_{1}A_{1} + … + x_{n}A_{n} = B among the columns, which leads to exactly the SAME system of scalar equations and the SAME augmented matrix as above. Thus we can also look at Gaussian reduction as a system from the point of view of columns. The question of linear independence of a set of column vectors A_{1},..A_{n} is the question of a NONTRIVIAL solution for the special homogeneous case B = 0. In this case the general solution x_{i} V_{i} + .. + x_{j}V_{j} = B = 0 means the vectors are independent iff there are NO FREE PARAMETERS, and they are dependent iff there ARE FREE PARAMETERS, in which case the V_{i} form a basis for the linear subspace of solutions of AX = 0, and the number of free parameters is the dimension of the solution space (consisting of all dependence relations among the columns). Moreover, the coefficients in each column vector V_{i} represent the coefficients of a nontrivial dependence relation: if V_{i} ^{tr} = [c1, c2, …, cn] ] then c1...
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This note was uploaded on 09/25/2009 for the course APMA 3080 taught by Professor Pindera during the Spring '09 term at UVA.

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