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Problem 13

Give an example of a pair of non-collinear vectors in \(R^{2}\). Then, show that the point \(\left(\mathrm{x}_{1}, \mathrm{x}_{2}\right)=(8,7)\) can be expressed as a linear combination of the non-collinear vectors.

Expert verified

A pair of non-collinear vectors in \(R^{2}\) could be \( \vec{a} = (1,0) \) and \( \vec{b} = (0,1) \). The point (8,7) can be expressed as a linear combination of these non-collinear vectors by finding scalars 's' and 't' such that \( s\vec{a} + t\vec{b} = (8,7) \). In this case, we have \(s = 8\) and \(t = 7\), so the linear combination is: (8, 7) = 8(1,0) + 7(0,1).

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