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n - dim Vector

n - dim Vector. n – dim vector and linear operations. I . The conception of n – dim vector. 1. Definition 1:. Column vector. Row vector. i th heft. Row vector of matrix A. 0 = ( 0,0,···,0 ). Column vector of matrix A. Null vector. Negative vector. 2. Def 2:.

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n - dim Vector

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  1. n-dimVector n –dim vector and linear operations I. The conception of n –dim vector 1. Definition 1: Column vector Row vector ith heft

  2. Row vector of matrix A 0 = ( 0,0,···,0 ) Column vector of matrix A Null vector Negative vector

  3. 2.Def 2: Let vectors II. Linear operations of n-dim vectors 1: 2:

  4. 3: Here the eight operation laws are all applicable, and you can find them in the textbook. Linear Dependence of Vector Set I. Linear dependence 1.Def 1:

  5. 2.Def 2: 0 (1)If there is only one vector in the set, and it is a null vector, we can say the set is linearly dependent; if it is not a null vector, then the set is linearly independent. (2) Two vectors are linearly dependent if and only if their corresponding quantities are proportioned. (3)Any set with null vector is linearly dependent. 3.Talk about the dependence of vector set.

  6. let O O O Solution： The determinant of coefficients is So, the system of equations has nonzero solutions, so there are numbers k1, k2, k3 which are not all 0,such that Solution ： That is

  7. O = O = O That is ： Coefficient determinant

  8. The Equivalent of Vector Sets 1.Def 1: Given two n-dim vector sets If every vectors in (I) can be linearly represented by vectors in (II),we say set (I) can be linearly represented by(II)； If sets（I）and（II）can be linearly represented by each other, then we say（I）and（II）are equivalent. The equivalent vector sets are reflexive, symmetric and transitive. e.g.1 Prove

  9. 0 O Determine About the Dependence and Some Important Results 1.Theory of the relations between linear dependence and combination. Prove

  10. 0 0 Prove Contradiction

  11. O So, the way to represent is unique.

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