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本文是討論4個n維向量問的一個問題,具體地來說,就是定理:設A=(a_1,a_2,…,a_n),B=(b_1,b_2,…,b_n),X=(x_1,x_2,…,x_n)和Y=(y_1,y_2,…,y_n)為4個非零的n維向量,其向量分適合 (1) a_ib_j+a_jb_i=x_iy_j+x_jy_i(i,j=1,2,…,n)之諸關係式:那麼A,B一定分别和X,Y或Y,X成比例,即必有二數λ≠0,μ≠0致A=λX,B=μY,或A=λY.B=μX。 證明:當n=1時,A=(a_1),B=(b_1),X=(x_1),Y=(y_1)。因題設A,B,X,Y均非零向量,故此時應為a_1b_1x_1y_1≠0,故A=λX,B=μY或A=σY,B=γX之4個異於零之數λ,μ,σ,γ之存在甚為顯明,此即示定理對於一維向量來講是成立的——實際上,由於(1)的原故,此時還顯然有λμ=1或σγ=1。今用數學歸納法假定定理對於n-1維向量而言是成立的,而來考察適合關係式(1)的4個n維向量A,B,X和Y。因A為非零向量,故它必至少有一個向量分
This article is to discuss a problem of four n-dimensional vectors. Specifically, the theorem: Let A = (a_1, a_2, ..., a_n), B = (b_1, b_2, ..., b_n), X = (x_1 ,x_2,...,x_n) and Y=(y_1,y_2,...,y_n) are four non-zero n-dimensional vectors whose vectors are suitable for (1) a_ib_j+a_jb_i=x_iy_j+x_jy_i(i,j=1, 2, ..., n) relations: then A, B must be proportional to X, Y or Y, X, respectively, that there must be two numbers λ ≠ 0, μ ≠ 0 to A = λX, B = μ Y, or A=λY.B=μX. Proof: When n=1, A=(a_1), B=(b_1), X=(x_1), Y=(y_1). Because the question is set to A, B, X, Y are non-zero vector, so this time should be a_1b_1x_1y_1≠0, so A = λX, B = μY or A = σY, B = γX different from the zero λ, μ The existence of σ and γ is quite obvious. This instant theorem holds true for a one-dimensional vector—in fact, due to the original reason of (1), there is obviously λμ=1 or σγ=1. Nowadays, the mathematical inductive method assumes that the theorem is true for the n-1 dimensional vector, and considers the four n-dimensional vectors A, B, X, and Y suitable for the relation (1). Since A is a non-zero vector, it must have at least one vector