Let b 2 4 2 Let W be the hyperplane b 2 a What is the dime

Let b = (2, 4, 2). Let W be the hyperplane b . [2] (a) What is the dimension of W? [4] (b) Find a basis for W. Make sure you justify the fact that it is a basis

Solution

a)

R3 has dimension 3

So , b perpendicular has dimension 2 since span(b) has dimension 1

b)

LEt, x be a vector in b perpendicular

So, x.b=0

x=(r,s,t)

So,

2r-4s+2t=0

r-2s+t=0

r=2s-t

HEnce,

x=(2s-t,s,t)=s(2,1,0)+t(-1,0,1)

So basis for W is {(2,1,0),(-1,01)}

Proof that this set of vectors is linearly independent

Let, p,q so that

p(2,1,0)+q(-1,0,1)=0

So, p=q=0

Hence thsi is a basis for W