differential equations math Please find the question in the

differential equations math

Please find the question in the the link below

http://imgur.com/aPOOCJK

Do each of the following: Prove that the set S of vectors described below determine a vector space. Find a basis for S and show that the vectors in your basis are linearly independent and Span S.

Solution

a. First we prove it forms a vector space

1. 0 belongs to the set

2. Let, (x,y,z),(p,q,r) belong to the set

(x,y,z)+(p,q,r) =(x+p,y+q,z+r)

So,

-4(x+p)-5(y+q)+z+r=-4x-5y+z+(-4p-5q+r)=0

HEnce, (x,y,z)+(p,q,r) belongs to the set.

3.

Let, (x,y,z) belong to the set and c be real number

c(x,y,z) =(cx,cy,cz)

So,

-4(cx)-5(cy)+cz=c(-4x-5y+z)=0

HEnce, c(x,y,z) belongs to the set.

Hence teh set is a vector space.

b)

For a vector (x,y,z) in teh set

-4x-5y+z=0

Hence, z=4x+5y

So ,(x,y,z)=(x,y,4x+5y)=x(1,0,4)+y(0,1,5)

But (x,y,z) was arbitrary vector. Hence all vector lie in span of

{(1,0,4),(0,1,5)}

So basis is

{(1,0,4),(0,1,5)}

c.

Basis has two vectors so dimension is 2.