Well explained ! But the professor said in the lecture that : C = c - (aTc/aTa)*a - (bTc/bTb)*b This will give me another answer which is wrong .. but I don't know why .. If we use this formula that means that we want a vector prep to a .. then take this vector and make it prep to b .. don't know why didn't work
since the short answer is already given, here is a deeper one. Let p and q be 2 dimensional vectors which are independent (but not orthogonal!). We know that p, q span the entire 2d space. Suppose, we want to express another 2d vector r = ap + bq. Can we say a = rTp/pTp and b = rTq/qTq? Draw it in a piece of paper and you will see we can't! we can write is such a fashion only if p and q are perpendicular.
2:21 "When you do the projection of a vector onto another, you have to divide by the length of in this case q1." Wouldn't it be dividing by the length squared? Or in other words, q1 dot q1? That's what the formula for Gram-Schmidt tells us: q2' = (q1 q1T b) / (q1T q1) where q1T q1 happens to be the length squared of q1.
You almost got it! For any vector a, aT a = |a|^2. In the case where the vector is q it is defined to have length 1. qT q = 1^2 = 1 An example might help. Look at the second term in Gram-Schmidt: a (aT b) / (aT a) = a(aT b) / |a|^2 = a/|a| * aT/|a| b but a/|a| = q1 and aT/|a| = q1T so a (aT b) / (aT a) = q1 (q1T b) I used the parentheses in the final line to show the meaning: project b along the direction of a to get a scalar. Multiply that by the direction of a. Hope this helps!
You might have found it out by yourself, but what she was doing is checking if A=QR, where Q came from the Gram-Schmidt, and R came from observation, (but she explained a general way to get R at the end). For b=2q1 + 3q2, it checked the second column of A (which is b) = the matrix Q multiply the second column of R (which is [2, 3, 0]), so it's b = R_{12}q_1 + R_{22}q_2 + R_{32}q_3. Hope this would be useful.
If you cannot follow the formula used here, I'd recommend going through this intro first: ruclips.net/video/rHonltF77zI/видео.html (Gram-Schmidt process by Khan Academy)
The ending explanation about how the entries in R can be seen/found in previous calculations was very helpful..
Thank you so much.
Excellent explanation and conclusion on QR relations
These videos are incredibly helpful in refreshing those ideas. It makes these concepts really stick in my mind.
Good understanding for R
Well explained ! But the professor said in the lecture that :
C = c - (aTc/aTa)*a - (bTc/bTb)*b
This will give me another answer which is wrong .. but I don't know why .. If we use this formula that means that we want a vector prep to a .. then take this vector and make it prep to b .. don't know why didn't work
the video uses the new vector a and b! if u use the new vector aTa and bTb will be 1.
since the short answer is already given, here is a deeper one. Let p and q be 2 dimensional vectors which are independent (but not orthogonal!). We know that p, q span the entire 2d space. Suppose, we want to express another 2d vector r = ap + bq. Can we say a = rTp/pTp and b = rTq/qTq? Draw it in a piece of paper and you will see we can't!
we can write is such a fashion only if p and q are perpendicular.
great eg, thank you Ana!
Excellent explanation
great great video. thank you Ana!
You are awesome!
2:21 "When you do the projection of a vector onto another, you have to divide by the length of in this case q1."
Wouldn't it be dividing by the length squared? Or in other words, q1 dot q1?
That's what the formula for Gram-Schmidt tells us: q2' = (q1 q1T b) / (q1T q1)
where q1T q1 happens to be the length squared of q1.
You almost got it! For any vector a, aT a = |a|^2. In the case where the vector is q it is defined to have length 1. qT q = 1^2 = 1
An example might help. Look at the second term in Gram-Schmidt:
a (aT b) / (aT a)
= a(aT b) / |a|^2
= a/|a| * aT/|a| b
but a/|a| = q1 and aT/|a| = q1T
so
a (aT b) / (aT a) = q1 (q1T b)
I used the parentheses in the final line to show the meaning: project b along the direction of a to get a scalar. Multiply that by the direction of a.
Hope this helps!
very clear!
Around 7:40, how do we go from 2 3 0 in the second column of R
to
b=2q1 + 3q2?
You might have found it out by yourself, but what she was doing is checking if A=QR, where Q came from the Gram-Schmidt, and R came from observation, (but she explained a general way to get R at the end).
For b=2q1 + 3q2, it checked the second column of A (which is b) = the matrix Q multiply the second column of R (which is [2, 3, 0]), so it's b = R_{12}q_1 + R_{22}q_2 + R_{32}q_3.
Hope this would be useful.
good one ana
If you cannot follow the formula used here, I'd recommend going through this intro first: ruclips.net/video/rHonltF77zI/видео.html (Gram-Schmidt process by Khan Academy)
thanks man, that really helps
great！
❤
only she can fix me