Determine which of the following is true or false, justify your answers. (a) If

Question

Determine which of the following is true or false, justify your answers. (a) If a set of vectors S is linearly dependent, then every vector can be represented as a linear combination of the other vectors in the set. (b) If the columns of an m times n matrix A span R^0m, then the equation Ax = b is consistent for each b in R^m. (c) A homogeneous system, Ax = 0, is always consistent. (d) The equation x= p + tv describes a line through v parallel to p. (e) The set Span {u, v} is always visualized as a plane through the origin.

Explanation / Answer

a)True ,given the set of vectors are linearly dependent

So those vectors can be written a1v1+a2v2+...+anvn =0 where

A1,a2,a3,...,an are constants and v1,v2,...,vn are vectors

Which inturn implies every vector can be represented as linear combination of other vectors

b)False ,as both A and b are span in Rm  ,but there are no variables so we can say they are consistent

c)True, yes there always exists a solution that is tribal one X=zero matrix

D)False,The given describes line through p and parallel to v.

e)True, the span of u,v can be viewed as plane as any point on that plane can be written as a*u+b*v where a,b are constants.

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