Question

let vector x = [a,b], vector y = [c,d]. Suppose that ac + bd = 0, and that a, b, c, and d are all nonzero. prove that vector x, y must be linearly independent.

Answer 1

If \(ac + bd=0\) then \(x\cdot y=|x||y|\cos \theta=0\). Since \(x\ne0,y\ne0\) \(\cos\theta=0\implies\theta=\cfrac\pi 2\). This means means \(x\) and \(y\) are orthogonal. For orthogonal vectors, there are no constants \(k_1,k_2\) such that \(k_1x+k_2y=0\). Therefore, \(x\) and \(y\) are linearly independent.

Answer 2

thank you so much

Answer 3

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