Find all ways of expressing 1 3 as a linear combination of 1

Find all ways of expressing (1, 3) as a linear combination of (1, 2), (2, 3), and (3, 4). Of course, use matrices to do so.

Solution

(1,3)=a(1,2)+b(2,3)+c(3,4);

Where a,b,c are constant.

(1,3)=(a+2b+3c,2a+3b+4c);

a+2b+3c=1........(1),

2a+3b+4c=3....…(2)

Solve equation (1) and (2),we get

Let c=k is any real number

a=3+k,b=-1-2k.

Suppose k=1,

(1,3)=4(1,2)+(-3)(2,3)+1(3,4).