91影视

Let V be a vector space over a field K. Let$v_1,v_2,v_3,...,v_n鈭�V$ , then if there exist scalars$a_1,a_2,a_3,...,a_n鈭�K$ , not all of them 0, such that

$a_1{v}_1+a_2{v}_{}+...a_n{v}_n=0$

Then the vectors $v_1,v_2,v_3,...,v_n$are called linearly dependent vectors.

Otherwise, linearly independent vectors

Let $\stackrel{鈫�}{u}\text{and}\stackrel{鈫�}{v}$are vectors that are linearly dependent. Then there exist scalars $a,b,c鈭�K,$not all of them zero, such that

role="math" localid="1664189159868" $$\begin{gathered} a\stackrel{鈫�}{u}+b\stackrel{鈫�}{v}+c\stackrel{鈫�}{w}=0 \\ 鈬�c\stackrel{鈫�}{w}=-a\stackrel{鈫�}{u}-b\stackrel{鈫�}{v} \\ 鈬�\stackrel{鈫�}{w}=-c^{-1}a\stackrel{鈫�}{u}-c^{-1}b\stackrel{鈫�}{v}\text{(where 'c'}鈮�\text{0)} \end{gathered}$$

This shows that the vector $\stackrel{鈫�}{w}$is a linear combination of vectors$\stackrel{鈫�}{u}\text{and}\stackrel{鈫�}{v}$ .

Since we have shown that$\stackrel{鈫�}{w}=-c^{-1}a\stackrel{鈫�}{u}-c^{-1}b\stackrel{鈫�}{v}\text{(where c}鈮�\text{0).}$

Hence, if vectors$\stackrel{鈫�}{u},\stackrel{鈫�}{v},\stackrel{鈫�}{w}$ are linearly dependent, then vector $\stackrel{鈫�}{w}$must be linear combination of vectors $\stackrel{鈫�}{u}\text{and}\stackrel{鈫�}{v}$.