# A sum of scalar multiple of two or more vectors (such as c_{1} u + c_{2} v + c_{3} w, where c_{i} are scalars) is called a linear combination of the vectors. Express \langle 4, -8 \rangle as a linear combination of i and j

A sum of scalar multiple of two or more vectors (such as $$\displaystyle{c}_{{{1}}}{u}+{c}_{{{2}}}{v}+{c}_{{{3}}}{w}$$, where $$\displaystyle{c}_{{{i}}}$$ are scalars) is called a linear combination of the vectors. Let $$\displaystyle{i}={\left\langle{1},{0}\right\rangle},{j}={\left\langle{0},{1}\right\rangle},{u}={\left\langle{1},{1}\right\rangle}$$, and $$\displaystyle{v}={\left\langle-{1},{1}\right\rangle}$$.
Express $$\displaystyle{\left\langle{4},-{8}\right\rangle}$$ as a linear combination of i and j (that is, find scalars $$\displaystyle{c}_{{{1}}}$$ and $$\displaystyle{c}_{{{2}}}$$ such that $$\displaystyle{\left\langle{4},-{8}\right\rangle}={c}_{{{1}}}{i}+{c}_{{{2}}}{j}$$).

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Cristiano Sears
Step 1
Given
$$\displaystyle{i}={\left\langle{1},{0}\right\rangle}$$ and $$\displaystyle{j}={\left\langle{0},{1}\right\rangle}$$
Step 2
$$\displaystyle{\left\langle{4},-{8}\right\rangle}={4}{i}-{8}{j}$$
so $$\displaystyle{c}_{{{1}}}={4}$$ and $$\displaystyle{c}_{{{2}}}=-{8}$$