Question

Let B = (a1, a2, a3) be an ordered basis of  R3 with a1 = (1, 0, -1), a2 = (1, 1, 1),  a3 = (1, 0, 0).

Write the vector v = (a, b, c) as  a linear combination of the basis vectors  from B.

Answer 1

Let $v = (a, b, c)=pa_1+qa_2+ra_3$ for some $p,q,r\in\mathbb R$ and find the coefficients $p,q,r$. We have the following

$(a, b, c)=p(1,0,-1)+q(1,1,1)+r(1,0,0)=(p+q+r,q, -p+q)$

So, $a=p+q+r,\ \ b=q$ and $c=-p+q$ . Therefore, $p=q-c=b-c$ and $r=a-p-q=a-(b-c)-b=a-2b+c$ .

Therefore, $v = (a, b, c)=(b-c)a_1+ba_2+(a-2b+c)a_3.$