Question

Let ${\mathbf{v}}_1$and ${\mathbf{v}}_2$be two nonzero position vectors in ${\mathrm{ℝ}}^2$that are not scalar multiples of each other. Explain why, given any vector $\mathbf{w}$in ${\mathrm{ℝ}}^2$, there are scalars $c_1$and $c_2$such that $\mathbf{w}=c_1{\mathbf{v}}_1+c_2{\mathbf{v}}_2$.

Short answer

$\text{There exists the scalars}{c}_1\text{and}{c}_2\text{such that}\mathbf{w}=c_1{\mathbf{v}}_1+c_2{\mathbf{v}}_2\text{.}$

Step-by-step solution

Step 1:Given information

.$v_{1}and{v}_2$be two nonzero position vectors

Step 2:Explaination

$\text{The objective is to explain given any vector}\mathbf{w}\text{in}{\mathrm{ℝ}}^2\text{, there are scalars}{c}_1\text{and}{c}_2\text{such that}$

$\mathbf{w}=c_1{\mathbf{v}}_1+c_2{\mathbf{v}}_2$

$\text{The vectors}{\mathbf{v}}_1\text{and}{\mathbf{v}}_2\text{in}{\mathrm{ℝ}}^2\text{are not the scalar multiple of each other. Thus, the vectors are}$

$\text{independent in}{\mathrm{ℝ}}^2\text{.}$

$\text{Thus, the vectors}{\mathbf{v}}_1\text{and}{\mathbf{v}}_2\text{in}{\mathrm{ℝ}}^2\text{behave as basis of}{\mathrm{ℝ}}^2$

$\text{The space}{\mathrm{ℝ}}^2\text{is spanned by the vectors}{\mathbf{v}}_1\text{and}{\mathbf{v}}_2\text{.}$

$\text{For any non-zero vector}\mathbf{w}\text{in}{\mathrm{ℝ}}^2\text{the vector}\mathbf{w}\text{is the linear combination of the vectors}{\mathbf{v}}_1\text{and}{\mathbf{v}}_2\text{.}$

$\text{Thus, there exists the scalars}{c}_1\text{and}{c}_2\text{such that}\mathbf{w}=c_1{\mathbf{v}}_1+c_2{\mathbf{v}}_2\text{.}$