APMA308-2%20Hmwk%203%20comments

Xntrexpressibleasalinearcombinationxxivixjvjofthebasic

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: A_{1} + c2 A_{2} + … + cn A_{n} = 0 is a nontrivial dependence relation among the columns. THUS GAUSSIAN REDUCTION CAN DETERMINE INDEPENDENCE OR DEPENDENCE OF A SET OF COLUMN VECTORS. Moreover, it tells you who to throw out to pare down to an independent spanning set of columns (every spanning set for a space can be shrunk down to a basis for that space): Throw out all the columns A_{k} that correspond to free parameters x_{k}, and the remaining A_{i} (corresponding to the leading one variables) will be independent: the dependence relations x_{1}A_{1} + … + x_{n}A_{n} = AX = 0 have X = [x_{1],..,x_{n}]^{tr} expressible as a linear combination X = x_{i} V_{i} + .. + x_{j}V_{j} of the basic solutions V_{k}, so the linear dependence relations x_{r}A_{r} + … + x_{s}A_{s} = 0 among the leading one columns have x_{k} = 0 for all the free parameter columns, and therefore are expressed in terms of the basic solutions as X = 0 V_{i} + .. + 0 V_{j}, in other words X = 0, so there are no nontrivial combinations of the leading‐one columns. This method of testing independence and spanning will be useful in the Webwork assignments. 1.5 # 24 Remember that the set of vectors orthogonal to W_{1}, .. w_{r} are the vectors v satisfying v.w_{i} = 0 for I = 1,..,r, which is a system of linear equations v_{1} a_{i1} + v_{2} a_{i2} +.. v_{n} a_{in} = 0, so you put the vectors w_{i} in...
View Full Document

This note was uploaded on 09/25/2009 for the course APMA 3080 taught by Professor Pindera during the Spring '09 term at UVA.

Ask a homework question - tutors are online