Question

If $\overrightarrow{v_1}and\overrightarrow{v_2}$ are 2 vectors, then $c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2}.\overrightarrow{v_2}$ is the linear combination of these 2 vectors where $c_1,c_2\stackrel{`}{\epsilon }R$.

• A. True
• B. False

Answer 1

The correct option is B False

Linear combination of n vectors as discussed is the sum of all vectors each scaled up or down by some scalar. Let’s say we have n vectors $\overrightarrow{v_1},\overrightarrow{v_2}........\overrightarrow{v_n}$, their linear combination will be $c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2}+....c_n\overrightarrow{v_n}$ where $c_1{c}_2.....c_n$ are all scalars (any real numbers). Here in each of the n terms each of the vector is multiplied by a scalar only and not by any other vector. Hence the word linear combination is used. So $c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2}$ will be the linear combination for vectors $\overrightarrow{v_1}and\overrightarrow{v_2}$ and not $c_1\overrightarrow{v_1}+c_2\overrightarrow{v_2}.\overrightarrow{v_2}$