6.1.1 Creating Linear Combinations

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Generalizations: 6.1 Linear Combination of Vectors

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Creating Linear Combinations

Find one linear combination of these vectors.

\begin{aligned} {\vec{a}}_{1} = [\begin{array}{c} - 1 \\ 1 \\ 1 \end{array}] & {\vec{a}}_{2} = [\begin{array}{c} - 1 \\ - 1 \\ 1 \end{array}] \end{aligned}

Solution. We will create a linear combination using the scalars $1$ and $2$ .

\begin{aligned} \vec{b} & = 1 \cdot {\vec{a}}_{1} + 2 \cdot {\vec{a}}_{2} \\ = 1 \cdot [\begin{array}{c} - 1 \\ 1 \\ 1 \end{array}] + 2 \cdot [\begin{array}{c} - 1 \\ - 1 \\ 1 \end{array}] & Scalar-vector multiplication \\ = [\begin{array}{c} - 1 \\ 1 \\ 1 \end{array}] + [\begin{array}{c} - 2 \\ - 2 \\ 2 \end{array}] & Vector-vector addition \\ = [\begin{array}{c} - 3 \\ - 1 \\ 3 \end{array}] \end{aligned}

Remark. This can be rewritten as a 10 Matrix-Vector Multiplication problem!

\vec{b} = [\begin{matrix} - 1 & - 1 \\ 1 & - 1 \\ 1 & 1 \end{matrix}] [\begin{matrix} 1 \\ 2 \end{matrix}]

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