→
u
=
→
w1
+
→
w2
,w1∈W and w2∈W⊥Let S[(x1,y1,z1)(x2,y2,z2)]=3x1x2+x2y1+x1y2+y1y2+z1z2 be an inner product,
W={(x,y,z)∈R3|x+z=0} a subspace and
→
u
=(1,0,0)
Find w1∈W, w2∈W⊥ (orthogonal in respect of S) such that
→
u
=
→
w
1+
→
w
2
Here's what I've done so far:
An orthogonal basis of W in respect of the inner product S is B={(1,0,−1),(−
1
4
,1,
1
4
)}
If w=(x,y,z)∈W⊥ then S[(x,y,z),(1,0,−1)]=0 and S[(x,y,z),(−
1
4
,1,
1
4
)]=0
So 3x+y−z=0 and
1
4
x+
3
4
y+
1
4
z=0⇒x+3y+z=0
Solving the above system we get that the basis for W⊥ in respect of S is B′={(1,−1,0),(0,0,1)}
I don't know how to proceed from here..
Also, are the
→
w
1,
→
w
2:
→
u
=
→
w
1+
→
w
2 unique?
=
→
w1
+
→
w2
,w1∈W and w2∈W⊥Let S[(x1,y1,z1)(x2,y2,z2)]=3x1x2+x2y1+x1y2+y1y2+z1z2 be an inner product,
W={(x,y,z)∈R3|x+z=0} a subspace and
→
u
=(1,0,0)
Find w1∈W, w2∈W⊥ (orthogonal in respect of S) such that
→
u
=
→
w
1+
→
w
2
Here's what I've done so far:
An orthogonal basis of W in respect of the inner product S is B={(1,0,−1),(−
1
4
,1,
1
4
)}
If w=(x,y,z)∈W⊥ then S[(x,y,z),(1,0,−1)]=0 and S[(x,y,z),(−
1
4
,1,
1
4
)]=0
So 3x+y−z=0 and
1
4
x+
3
4
y+
1
4
z=0⇒x+3y+z=0
Solving the above system we get that the basis for W⊥ in respect of S is B′={(1,−1,0),(0,0,1)}
I don't know how to proceed from here..
Also, are the
→
w
1,
→
w
2:
→
u
=
→
w
1+
→
w
2 unique?
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