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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.

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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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## Step 1: Find a pair of non-collinear vectors in R^2.

Choose arbitrary vectors that are not scalar multiples of each other. One such possible pair could be: $$\vec{a} = (1,0)$$ and $$\vec{b} = (0,1)$$. Since their directions are different, these two vectors are non-collinear.

## Step 2: Express the point (8,7) as a linear combination of the non-collinear vectors

Now we want to find scalars 's' and 't' such that: $$s\vec{a} + t\vec{b} = (8,7)$$ Using the vectors defined in step 1, the equation becomes: $$s(1, 0) + t(0, 1) = (8, 7)$$ This simplifies to two separate equations: $$s = 8$$ $$t = 7$$ Since we found values for 's' and 't' that satisfy the equation, we can express (8,7) as a linear combination of the non-collinear vectors $$\vec{a}$$ and $$\vec{b}$$: $$(8, 7) = 8(1,0) +7(0,1)$$

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