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Find scalars that satisfy a given vector equation

Let

    \[ A = (1,2), \qquad B = (2,-4), \qquad C = (2,-3). \]

Find x,y \in \mathbb{R} such that C = xA + yB. How many pairs (x,y) are there that satisfy this equation?


First, A and B are linearly independent since c_1 A + c_2 B = 0 implies

    \[ c_1 + 2c_2 = 0 \quad \text{and} \quad 2c_1 - 4c_2 = 0 \]

which only has the solution c_1 = c_2 = 0. Therefore, A and B form a basis for \mathbb{R}^2 so there is a unique pair (x,y) such that C = xA + yB. To compute them we have the two equations

    \[ x+2y = 2 \qquad \text{and} \qquad 2x-4y = -3. \]

From the first equation we have x = 2 - 2y. Therefore from the second equation we have

    \[ 2(2-2y) - 4y = -3 \quad \implies \quad y = \frac{7}{8} \quad \implies \quad x = \frac{1}{4}. \]

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