APMA308-2%20Hmwk%203%20comments

Butifwewanttotestaset

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Ask a homework question - tutors are online