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Finding an orthonormal basis of the subspace using Gram-Schmidt method
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Apply the Gram-Schmidt orthonormalization process to the vectors $[1,3,2]^T$ and $[1,0,1]^T$ in order to get an orthonormal basis of the subspace that they span.
My Try:
I took $u_1=[1,3,2]^T,u_2=[1,0,1]^T$ and used the formula $V_1=dfracu_1u_1$ and $V_2=dfracu_2-langle u_2V_1rangle V_1$
I got $V_1=left[dfrac1sqrt14,dfrac3sqrt14,dfrac2sqrt14right]^T$
Now, I don't whether I am doing correct for $V_2$ or not. But this is what I did.
$$langle u_2V_1rangle V_1=left<(1,0,1)^Tleft(dfrac1sqrt14,dfrac3sqrt14,dfrac2sqrt14right)^Tright>^Tleft<dfrac1sqrt14,dfrac3sqrt14,dfrac2sqrt14right>^T=left<frac314,frac914,frac614right>^T$$
and $$u_2-langle u_2V_1rangle V_1=left<frac1114,frac-914,frac414right>^T$$
$$V_2=left<frac11sqrt218,frac-9sqrt218,frac4sqrt218right>^T$$
Is my $V_2$ correct?
linear-algebra gram-schmidt
edited Jul 22 at 23:01
mechanodroid
22.2k52041
asked Jul 22 at 22:36
philip
1158
add a comment |Â
up vote
1
down vote
favorite
Apply the Gram-Schmidt orthonormalization process to the vectors $[1,3,2]^T$ and $[1,0,1]^T$ in order to get an orthonormal basis of the subspace that they span.
My Try:
I took $u_1=[1,3,2]^T,u_2=[1,0,1]^T$ and used the formula $V_1=dfracu_1u_1$ and $V_2=dfracu_2-langle u_2V_1rangle V_1$
I got $V_1=left[dfrac1sqrt14,dfrac3sqrt14,dfrac2sqrt14right]^T$
Now, I don't whether I am doing correct for $V_2$ or not. But this is what I did.
$$langle u_2V_1rangle V_1=left<(1,0,1)^Tleft(dfrac1sqrt14,dfrac3sqrt14,dfrac2sqrt14right)^Tright>^Tleft<dfrac1sqrt14,dfrac3sqrt14,dfrac2sqrt14right>^T=left<frac314,frac914,frac614right>^T$$
and $$u_2-langle u_2V_1rangle V_1=left<frac1114,frac-914,frac414right>^T$$
$$V_2=left<frac11sqrt218,frac-9sqrt218,frac4sqrt218right>^T$$
Is my $V_2$ correct?
linear-algebra gram-schmidt
edited Jul 22 at 23:01
mechanodroid
22.2k52041
asked Jul 22 at 22:36
philip
1158
1
1) are your two solution vectors unit-length? 2) are they orthogonal? 3) do they span the correct subspace (easiest check is to see if they’re orthogonal to the cross product of the two input vectors)?
– user7530
Jul 22 at 22:42
@user7530 You mean do I need to do dot product of $V_1cdot V_2$ to check those?
– philip
Jul 22 at 22:43
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Apply the Gram-Schmidt orthonormalization process to the vectors $[1,3,2]^T$ and $[1,0,1]^T$ in order to get an orthonormal basis of the subspace that they span.
My Try:
I took $u_1=[1,3,2]^T,u_2=[1,0,1]^T$ and used the formula $V_1=dfracu_1u_1$ and $V_2=dfracu_2-langle u_2V_1rangle V_1$
I got $V_1=left[dfrac1sqrt14,dfrac3sqrt14,dfrac2sqrt14right]^T$
Now, I don't whether I am doing correct for $V_2$ or not. But this is what I did.
$$langle u_2V_1rangle V_1=left<(1,0,1)^Tleft(dfrac1sqrt14,dfrac3sqrt14,dfrac2sqrt14right)^Tright>^Tleft<dfrac1sqrt14,dfrac3sqrt14,dfrac2sqrt14right>^T=left<frac314,frac914,frac614right>^T$$
and $$u_2-langle u_2V_1rangle V_1=left<frac1114,frac-914,frac414right>^T$$
$$V_2=left<frac11sqrt218,frac-9sqrt218,frac4sqrt218right>^T$$
Is my $V_2$ correct?
linear-algebra gram-schmidt
edited Jul 22 at 23:01
mechanodroid
22.2k52041
asked Jul 22 at 22:36
philip
1158
Apply the Gram-Schmidt orthonormalization process to the vectors $[1,3,2]^T$ and $[1,0,1]^T$ in order to get an orthonormal basis of the subspace that they span.
My Try:
I took $u_1=[1,3,2]^T,u_2=[1,0,1]^T$ and used the formula $V_1=dfracu_1u_1$ and $V_2=dfracu_2-langle u_2V_1rangle V_1$
I got $V_1=left[dfrac1sqrt14,dfrac3sqrt14,dfrac2sqrt14right]^T$
Now, I don't whether I am doing correct for $V_2$ or not. But this is what I did.
$$langle u_2V_1rangle V_1=left<(1,0,1)^Tleft(dfrac1sqrt14,dfrac3sqrt14,dfrac2sqrt14right)^Tright>^Tleft<dfrac1sqrt14,dfrac3sqrt14,dfrac2sqrt14right>^T=left<frac314,frac914,frac614right>^T$$
and $$u_2-langle u_2V_1rangle V_1=left<frac1114,frac-914,frac414right>^T$$
$$V_2=left<frac11sqrt218,frac-9sqrt218,frac4sqrt218right>^T$$
Is my $V_2$ correct?
linear-algebra gram-schmidt
edited Jul 22 at 23:01
mechanodroid
22.2k52041
asked Jul 22 at 22:36
philip
1158
edited Jul 22 at 23:01
mechanodroid
22.2k52041
edited Jul 22 at 23:01
mechanodroid
22.2k52041
edited Jul 22 at 23:01
mechanodroid
22.2k52041
22.2k52041
asked Jul 22 at 22:36
philip
1158
asked Jul 22 at 22:36
philip
1158
asked Jul 22 at 22:36
philip
1158
1158
1
1) are your two solution vectors unit-length? 2) are they orthogonal? 3) do they span the correct subspace (easiest check is to see if they’re orthogonal to the cross product of the two input vectors)?
– user7530
Jul 22 at 22:42
@user7530 You mean do I need to do dot product of $V_1cdot V_2$ to check those?
– philip
Jul 22 at 22:43
add a comment |Â
1
1) are your two solution vectors unit-length? 2) are they orthogonal? 3) do they span the correct subspace (easiest check is to see if they’re orthogonal to the cross product of the two input vectors)?
– user7530
Jul 22 at 22:42
@user7530 You mean do I need to do dot product of $V_1cdot V_2$ to check those?
– philip
Jul 22 at 22:43
1
1
1) are your two solution vectors unit-length? 2) are they orthogonal? 3) do they span the correct subspace (easiest check is to see if they’re orthogonal to the cross product of the two input vectors)?
– user7530
Jul 22 at 22:42
1) are your two solution vectors unit-length? 2) are they orthogonal? 3) do they span the correct subspace (easiest check is to see if they’re orthogonal to the cross product of the two input vectors)?
– user7530
Jul 22 at 22:42
@user7530 You mean do I need to do dot product of $V_1cdot V_2$ to check those?
– philip
Jul 22 at 22:43
@user7530 You mean do I need to do dot product of $V_1cdot V_2$ to check those?
– philip
Jul 22 at 22:43
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
1
down vote
accepted
You made a small mistake, it should be:
$$u_2-langle u_2,V_1rangle V_1=left<frac1114,frac-914,fraccolorred814right>^T$$
and then
$$V_2 = left<frac11sqrt266,frac-9sqrt266,frac8sqrt266right>^T$$
Now you can check that $|V_1| = |V_2| = 1$ and $langle V_1, V_2rangle = 0$ so they are correct.
answered Jul 22 at 22:59
mechanodroid
22.2k52041
Thanks for the verification and for finding the mistake.
– philip
Jul 22 at 23:00
then how to verify if I had $V_1,V_2,V_3$
– philip
Jul 22 at 23:31
@philip $|V_1| = |V_2| = |V_3| = 1$ and $langle V_1, V_2rangle = langle V_1, V_3rangle = langle V_2, V_3rangle = 0$.
– mechanodroid
Jul 22 at 23:39
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
You made a small mistake, it should be:
$$u_2-langle u_2,V_1rangle V_1=left<frac1114,frac-914,fraccolorred814right>^T$$
and then
$$V_2 = left<frac11sqrt266,frac-9sqrt266,frac8sqrt266right>^T$$
Now you can check that $|V_1| = |V_2| = 1$ and $langle V_1, V_2rangle = 0$ so they are correct.
answered Jul 22 at 22:59
mechanodroid
22.2k52041
Thanks for the verification and for finding the mistake.
– philip
Jul 22 at 23:00
then how to verify if I had $V_1,V_2,V_3$
– philip
Jul 22 at 23:31
@philip $|V_1| = |V_2| = |V_3| = 1$ and $langle V_1, V_2rangle = langle V_1, V_3rangle = langle V_2, V_3rangle = 0$.
– mechanodroid
Jul 22 at 23:39
add a comment |Â
up vote
1
down vote
accepted
You made a small mistake, it should be:
$$u_2-langle u_2,V_1rangle V_1=left<frac1114,frac-914,fraccolorred814right>^T$$
and then
$$V_2 = left<frac11sqrt266,frac-9sqrt266,frac8sqrt266right>^T$$
Now you can check that $|V_1| = |V_2| = 1$ and $langle V_1, V_2rangle = 0$ so they are correct.
answered Jul 22 at 22:59
mechanodroid
22.2k52041
Thanks for the verification and for finding the mistake.
– philip
Jul 22 at 23:00
then how to verify if I had $V_1,V_2,V_3$
– philip
Jul 22 at 23:31
@philip $|V_1| = |V_2| = |V_3| = 1$ and $langle V_1, V_2rangle = langle V_1, V_3rangle = langle V_2, V_3rangle = 0$.
– mechanodroid
Jul 22 at 23:39
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
You made a small mistake, it should be:
$$u_2-langle u_2,V_1rangle V_1=left<frac1114,frac-914,fraccolorred814right>^T$$
and then
$$V_2 = left<frac11sqrt266,frac-9sqrt266,frac8sqrt266right>^T$$
Now you can check that $|V_1| = |V_2| = 1$ and $langle V_1, V_2rangle = 0$ so they are correct.
answered Jul 22 at 22:59
mechanodroid
22.2k52041
You made a small mistake, it should be:
$$u_2-langle u_2,V_1rangle V_1=left<frac1114,frac-914,fraccolorred814right>^T$$
and then
$$V_2 = left<frac11sqrt266,frac-9sqrt266,frac8sqrt266right>^T$$
Now you can check that $|V_1| = |V_2| = 1$ and $langle V_1, V_2rangle = 0$ so they are correct.
answered Jul 22 at 22:59
mechanodroid
22.2k52041
answered Jul 22 at 22:59
mechanodroid
22.2k52041
answered Jul 22 at 22:59
mechanodroid
22.2k52041
answered Jul 22 at 22:59
mechanodroid
22.2k52041
22.2k52041
Thanks for the verification and for finding the mistake.
– philip
Jul 22 at 23:00
then how to verify if I had $V_1,V_2,V_3$
– philip
Jul 22 at 23:31
@philip $|V_1| = |V_2| = |V_3| = 1$ and $langle V_1, V_2rangle = langle V_1, V_3rangle = langle V_2, V_3rangle = 0$.
– mechanodroid
Jul 22 at 23:39
add a comment |Â
Thanks for the verification and for finding the mistake.
– philip
Jul 22 at 23:00
then how to verify if I had $V_1,V_2,V_3$
– philip
Jul 22 at 23:31
@philip $|V_1| = |V_2| = |V_3| = 1$ and $langle V_1, V_2rangle = langle V_1, V_3rangle = langle V_2, V_3rangle = 0$.
– mechanodroid
Jul 22 at 23:39
Thanks for the verification and for finding the mistake.
– philip
Jul 22 at 23:00
Thanks for the verification and for finding the mistake.
– philip
Jul 22 at 23:00
then how to verify if I had $V_1,V_2,V_3$
– philip
Jul 22 at 23:31
then how to verify if I had $V_1,V_2,V_3$
– philip
Jul 22 at 23:31
@philip $|V_1| = |V_2| = |V_3| = 1$ and $langle V_1, V_2rangle = langle V_1, V_3rangle = langle V_2, V_3rangle = 0$.
– mechanodroid
Jul 22 at 23:39
@philip $|V_1| = |V_2| = |V_3| = 1$ and $langle V_1, V_2rangle = langle V_1, V_3rangle = langle V_2, V_3rangle = 0$.
– mechanodroid
Jul 22 at 23:39
add a comment |Â
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1
1) are your two solution vectors unit-length? 2) are they orthogonal? 3) do they span the correct subspace (easiest check is to see if they’re orthogonal to the cross product of the two input vectors)?
– user7530
Jul 22 at 22:42
@user7530 You mean do I need to do dot product of $V_1cdot V_2$ to check those?
– philip
Jul 22 at 22:43