Inner product of distance between two vectors
We can define a 'distance' between two points $P = (x_1,y_1)$ and $Q=(x_2,y_2)$ of the plane by $d(P,Q) = |x_2 - x_1| + |y_2 - y_1|$. Verify if the sentence below is a inner product in the plane.
$$langle(x_1,y_1),(x_2,y_2)rangle = d(P,Q)$$
UPDATE: What i've already made
Positivity
$langle P,Prangle geq 0 $
The distance from a point to a point is 0. [check]
Symmetry
$ langle P,Q rangle = langle Q,P rangle$
The distance from a point P to a point Q is equal to distance from a point Q to the point Q.[check]
Bilinearity
$ langlelambda P,Qrangle = lambdalangle P,Qrangle \
langle(lambda x_1,lambda y_1),(x_2,y_2)rangle = |x_2 - lambda x_1| + | y_2 - lambda y_1 |\$
I'm stuck here, i don't know how get out with this.
linear-algebra
asked Nov 29 '18 at 20:37
Messias TayllanMessias Tayllan
84
|
show 1 more comment
We can define a 'distance' between two points $P = (x_1,y_1)$ and $Q=(x_2,y_2)$ of the plane by $d(P,Q) = |x_2 - x_1| + |y_2 - y_1|$. Verify if the sentence below is a inner product in the plane.
$$langle(x_1,y_1),(x_2,y_2)rangle = d(P,Q)$$
UPDATE: What i've already made
Positivity
$langle P,Prangle geq 0 $
The distance from a point to a point is 0. [check]
Symmetry
$ langle P,Q rangle = langle Q,P rangle$
The distance from a point P to a point Q is equal to distance from a point Q to the point Q.[check]
Bilinearity
$ langlelambda P,Qrangle = lambdalangle P,Qrangle \
langle(lambda x_1,lambda y_1),(x_2,y_2)rangle = |x_2 - lambda x_1| + | y_2 - lambda y_1 |\$
I'm stuck here, i don't know how get out with this.
linear-algebra
asked Nov 29 '18 at 20:37
Messias TayllanMessias Tayllan
84
1
What are the properties of an inner product? How would you verify them in this case?
– rubikscube09
Nov 29 '18 at 20:43
What have you tried? Show us your work and maybe we can help you.
– Tito Eliatron
Nov 29 '18 at 20:54
Bilinearity,symmetry and positivity. I've tried and got the symmetry and positivity, but the linearity (even making each linearity separated) i do not.
– Messias Tayllan
Nov 29 '18 at 21:52
Have you considered that it might not be an inner product? Try a few examples and see if bilinearity holds for those examples.
– Kevin Long
Nov 29 '18 at 22:23
Yeah, the answer from J.G. helped me. I did not try before because the answer from my professor was that is a inner product, but i was confusing how do this.
– Messias Tayllan
Nov 29 '18 at 22:27
|
show 1 more comment
We can define a 'distance' between two points $P = (x_1,y_1)$ and $Q=(x_2,y_2)$ of the plane by $d(P,Q) = |x_2 - x_1| + |y_2 - y_1|$. Verify if the sentence below is a inner product in the plane.
$$langle(x_1,y_1),(x_2,y_2)rangle = d(P,Q)$$
UPDATE: What i've already made
Positivity
$langle P,Prangle geq 0 $
The distance from a point to a point is 0. [check]
Symmetry
$ langle P,Q rangle = langle Q,P rangle$
The distance from a point P to a point Q is equal to distance from a point Q to the point Q.[check]
Bilinearity
$ langlelambda P,Qrangle = lambdalangle P,Qrangle \
langle(lambda x_1,lambda y_1),(x_2,y_2)rangle = |x_2 - lambda x_1| + | y_2 - lambda y_1 |\$
I'm stuck here, i don't know how get out with this.
linear-algebra
asked Nov 29 '18 at 20:37
Messias TayllanMessias Tayllan
84
We can define a 'distance' between two points $P = (x_1,y_1)$ and $Q=(x_2,y_2)$ of the plane by $d(P,Q) = |x_2 - x_1| + |y_2 - y_1|$. Verify if the sentence below is a inner product in the plane.
$$langle(x_1,y_1),(x_2,y_2)rangle = d(P,Q)$$
UPDATE: What i've already made
Positivity
$langle P,Prangle geq 0 $
The distance from a point to a point is 0. [check]
Symmetry
$ langle P,Q rangle = langle Q,P rangle$
The distance from a point P to a point Q is equal to distance from a point Q to the point Q.[check]
Bilinearity
$ langlelambda P,Qrangle = lambdalangle P,Qrangle \
langle(lambda x_1,lambda y_1),(x_2,y_2)rangle = |x_2 - lambda x_1| + | y_2 - lambda y_1 |\$
I'm stuck here, i don't know how get out with this.
linear-algebra
linear-algebra
asked Nov 29 '18 at 20:37
Messias TayllanMessias Tayllan
84
asked Nov 29 '18 at 20:37
Messias TayllanMessias Tayllan
84
edited Nov 29 '18 at 22:15
Messias Tayllan
asked Nov 29 '18 at 20:37
Messias TayllanMessias Tayllan
84
asked Nov 29 '18 at 20:37
Messias TayllanMessias Tayllan
84
asked Nov 29 '18 at 20:37
Messias TayllanMessias Tayllan
84
84
1
What are the properties of an inner product? How would you verify them in this case?
– rubikscube09
Nov 29 '18 at 20:43
What have you tried? Show us your work and maybe we can help you.
– Tito Eliatron
Nov 29 '18 at 20:54
Bilinearity,symmetry and positivity. I've tried and got the symmetry and positivity, but the linearity (even making each linearity separated) i do not.
– Messias Tayllan
Nov 29 '18 at 21:52
Have you considered that it might not be an inner product? Try a few examples and see if bilinearity holds for those examples.
– Kevin Long
Nov 29 '18 at 22:23
Yeah, the answer from J.G. helped me. I did not try before because the answer from my professor was that is a inner product, but i was confusing how do this.
– Messias Tayllan
Nov 29 '18 at 22:27
|
show 1 more comment
1
What are the properties of an inner product? How would you verify them in this case?
– rubikscube09
Nov 29 '18 at 20:43
What have you tried? Show us your work and maybe we can help you.
– Tito Eliatron
Nov 29 '18 at 20:54
Bilinearity,symmetry and positivity. I've tried and got the symmetry and positivity, but the linearity (even making each linearity separated) i do not.
– Messias Tayllan
Nov 29 '18 at 21:52
Have you considered that it might not be an inner product? Try a few examples and see if bilinearity holds for those examples.
– Kevin Long
Nov 29 '18 at 22:23
Yeah, the answer from J.G. helped me. I did not try before because the answer from my professor was that is a inner product, but i was confusing how do this.
– Messias Tayllan
Nov 29 '18 at 22:27
1
1
What are the properties of an inner product? How would you verify them in this case?
– rubikscube09
Nov 29 '18 at 20:43
What are the properties of an inner product? How would you verify them in this case?
– rubikscube09
Nov 29 '18 at 20:43
What have you tried? Show us your work and maybe we can help you.
– Tito Eliatron
Nov 29 '18 at 20:54
What have you tried? Show us your work and maybe we can help you.
– Tito Eliatron
Nov 29 '18 at 20:54
Bilinearity,symmetry and positivity. I've tried and got the symmetry and positivity, but the linearity (even making each linearity separated) i do not.
– Messias Tayllan
Nov 29 '18 at 21:52
Bilinearity,symmetry and positivity. I've tried and got the symmetry and positivity, but the linearity (even making each linearity separated) i do not.
– Messias Tayllan
Nov 29 '18 at 21:52
Have you considered that it might not be an inner product? Try a few examples and see if bilinearity holds for those examples.
– Kevin Long
Nov 29 '18 at 22:23
Have you considered that it might not be an inner product? Try a few examples and see if bilinearity holds for those examples.
– Kevin Long
Nov 29 '18 at 22:23
Yeah, the answer from J.G. helped me. I did not try before because the answer from my professor was that is a inner product, but i was confusing how do this.
– Messias Tayllan
Nov 29 '18 at 22:27
Yeah, the answer from J.G. helped me. I did not try before because the answer from my professor was that is a inner product, but i was confusing how do this.
– Messias Tayllan
Nov 29 '18 at 22:27
|
show 1 more comment
1 Answer
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It's not an inner product because it isn't linear. For example, the choice $P=Q=(1,,0)$ implies $d(P,,kQ)=|k-1|$, which for $kne 1$ differs from $kd(P,,Q)=0$.
answered Nov 29 '18 at 22:08
J.G.J.G.
23.4k22137
add a comment |
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It's not an inner product because it isn't linear. For example, the choice $P=Q=(1,,0)$ implies $d(P,,kQ)=|k-1|$, which for $kne 1$ differs from $kd(P,,Q)=0$.
answered Nov 29 '18 at 22:08
J.G.J.G.
23.4k22137
add a comment |
It's not an inner product because it isn't linear. For example, the choice $P=Q=(1,,0)$ implies $d(P,,kQ)=|k-1|$, which for $kne 1$ differs from $kd(P,,Q)=0$.
answered Nov 29 '18 at 22:08
J.G.J.G.
23.4k22137
add a comment |
It's not an inner product because it isn't linear. For example, the choice $P=Q=(1,,0)$ implies $d(P,,kQ)=|k-1|$, which for $kne 1$ differs from $kd(P,,Q)=0$.
answered Nov 29 '18 at 22:08
J.G.J.G.
23.4k22137
It's not an inner product because it isn't linear. For example, the choice $P=Q=(1,,0)$ implies $d(P,,kQ)=|k-1|$, which for $kne 1$ differs from $kd(P,,Q)=0$.
answered Nov 29 '18 at 22:08
J.G.J.G.
23.4k22137
answered Nov 29 '18 at 22:08
J.G.J.G.
23.4k22137
answered Nov 29 '18 at 22:08
J.G.J.G.
23.4k22137
answered Nov 29 '18 at 22:08
J.G.J.G.
23.4k22137
23.4k22137
add a comment |
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1
What are the properties of an inner product? How would you verify them in this case?
– rubikscube09
Nov 29 '18 at 20:43
What have you tried? Show us your work and maybe we can help you.
– Tito Eliatron
Nov 29 '18 at 20:54
Bilinearity,symmetry and positivity. I've tried and got the symmetry and positivity, but the linearity (even making each linearity separated) i do not.
– Messias Tayllan
Nov 29 '18 at 21:52
Have you considered that it might not be an inner product? Try a few examples and see if bilinearity holds for those examples.
– Kevin Long
Nov 29 '18 at 22:23
Yeah, the answer from J.G. helped me. I did not try before because the answer from my professor was that is a inner product, but i was confusing how do this.
– Messias Tayllan
Nov 29 '18 at 22:27