1 Section 1.7. Linear Independence Denition 1 Asetofpvectors~u ;~u ;:::;~u in Rm is called linearly independent if the vector 1 2 p equation ~ x1~u1 + x2~u2 + ::: + xp~up = 0 has only the trivial solution x = 0; i = 1;2;:::;p: Otherwise, the set is called linearly depen- i dent; the coe¢ cients x1;:::;xn are called a linear relation. Example 2 Anyonesingle vector ~u is always independent. two vectors ~u ;~u are dependent 1 2 i¤ ~u1 = ~u2. The notation of linear dependence (or independence) is closely related to ...
9.1 Linear Independence Performance Criterion: 9. (a) Determine whether a set v ;v ;:::;v of vectors is a linearly independent 1 2 k or linearly dependent. If the vectors are linearly dependent, (1) give a non- trivial linear combination of them that equals the zero vector, (2) give any one as a linear combination of the others, when possible. Suppose that we are trying to create a set S of vectors that spans R3. We might begin with one vector, say −3 1 u1 = , in S. We know by now that the span of this single ...
Linear Independence Linear Independence. Denition. Let v ,v ,...,v ∈ Rn be a set of vectors. 1 2 k • The vectors are linearly dependent if there exist scalars λ1,λ2,...,λk ∈ R, not all zero, such that λ v +λ v +···+λ v =0. 1 1 2 2 k k • The vectors are linearly independent if λ v + ··· + λ v = 0 for scalars 1 1 k k λ1,...,λk implies that λ1 = λ2 = ··· = λk ...
PDF | 0.07 MB | Free Download
