Let u v w be linearly independent vectors in Rn Suppose c1 c

Let u, v, w be linearly independent vectors in R^n. Suppose c_1, c_2, c_3, d_1, d_2, d_3 are numbers such that c_1 v + c_2 w + c_3 u = d_1 v + d_2 w + d_3 u. True or false: this happens if and only if c_1 = d_1, c_2 = d_2, and c_3 = d_3. Justify your answer.

Solution

If u,v,w are linearly independent vectors in Rⁿ, then no linear combination of these vectors can be zero unless all the scalar coefficients are zero. Thus, is c₁v + c₂w +c₃u = d₁v+ d₂w +d₃u, then we have (c₁v + c₂w +c₃u) – (d₁v+ d₂w +d₃u) = 0 or, (c₁ – d₁)v + (c₂-d₂) w + (c₃-d₃)u = 0. Therefore, c₁ – d₁ =0, c₂-d₂= 0 and c₃-d₃ =0. Hence c₁ = d₁ , c₂ = d₂ and c₃ =d₃.