Q .18. - Calculus

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

Step-by-Step Solution

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 1:Given information

.$v_1 and v_2$ be two nonzero position vectors

Step 2:Explaination

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

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

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

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

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

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

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

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