Chapter 8, Problem 1SE

To determine

To state: The statement "Given $v_1,v_2,\dots \dots ,v_p$ in ${ℝ}^n$ and scalars $c_1,\mathrm{...},c_p$ , an affine combination of $v_1,v_2,\dots .,v_p$ is a linear combination $c_1{v}_1+\dots \dots +c_p{v}_p$ such that the weights satisfy $c_1+\dots \mathrm{..}+c_p=1$ . "$ is true or false.

Answer to Problem 1SE

The statement "Given $v_1,v_2,\dots \dots ,v_p$ in ${ℝ}^n$ and scalars $c_1,\mathrm{...},c_p$ , an affine combination of $v_1,v_2,\dots .,v_p$ is a linear combination $c_1{v}_1+\dots \dots +c_p{v}_p$ such that the weights satisfy $c_1+\dots \mathrm{..}+c_p=1$ . "$ is true

Explanation of Solution

Given information:

The statement is "Given $v_1,v_2,\dots \dots ,v_p$ in ${ℝ}^n$ and scalars $c_1,\mathrm{...},c_p$ , an affine combination of $v_1,v_2,\dots .,v_p$ is a linear combination $c_1{v}_1+\dots \dots +c_p{v}_p$ such that the weights satisfy $c_1+\dots \mathrm{..}+c_p=1$ . "

Proof:

An affine combination of vectors is a special kind of linear combination. Given vectors $v_1,v_2,\dots .,v_p$ in ${ℝ}^n$ and scalars $c_1,\dots .,c_p$ , an affine combination of $v_1,v_2,\dots \dots ,v_p$ is a linear combination $c_1{v}_1+\dots \dots +c_p{v}_p$ such that the weights satisfy $c_1+\dots \dots +c_p=1$ .

From the definition of affine combination, it is safe to say that, An affine combination of vectors is a special kind of linear combination.

Given vectors $v_1,v_2,\dots \mathrm{..},v_p$ in ${ℝ}^n$ and scalars $c_1,\dots \mathrm{..},c_p$ , an affine combination of $v_1,v_2,\dots \dots ,v_p$ is a linear combination $c_1{v}_1+\dots \dots +c_p{v}_p$ such that the weights satisfy $c_1+\dots .+c_p=1$ .

Hence, given $v_1,v_2,\dots .,v_p$ in ${ℝ}^n$ and scalars $c_1,\dots .,c_p$ , an affine combination of $v_1,v_2,\dots .,v_p$ is a linear combination $c_1{v}_1+\dots .+c_p{v}_p$ such that the weights satisfy $c_1+\dots .+c_p=1$ .

Therefore, the given statement is true.