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# Projections onto subspaces

Projections onto subspaces. Created by Sal Khan.

## Want to join the conversation?

• at sal refers to another video, can someone please refer me to it. thanks,
also... why not add an embedded comment with a link to the video if your'e going to reference it?
• At the projection is calculated as [-27/13, -18/13]. But the projection vector has a positive horizontal component (it's pointing to the right). Am I missing something? I'm assuming that vector is w.r.t to the original space (vs. the null+row space) since the projection is calculated using vectors from that space.
• The projection is [27/13, -18/13], not [-27/13, -18/13]. The correct projection does point to the right along the rowspace of A.
• How can vector be placed anywhere and still be considered the same vector like @ ? then why would the solution set @ be a different vector? isn't that just moving the null space by 3 to the x-direction..?
(1 vote)
• I think it's true that you can draw a vector anywhere (because a vector just has magnitude and direction). However, when you want to use vectors to describe points in a vector space (i.e. when you want your vectors to be position vectors), you need the vector space to have an origin and you need to describe all the points with respect to that origin.

This means, for example, that when you are using the sum of two vectors to describe a point, one of the vectors has to start from the origin (and the other vector would be drawn head-to-tail with it because that is how vectors add). This also applies to describing lines in a vector space.
• what is the difference between null space and complement?
(1 vote)
• If you mean the Orthogonal Complement (https://www.khanacademy.org/math/linear-algebra/alternate_bases/othogonal_complements/v/linear-algebra-orthogonal-complements), then they are closely related, but are a different set of subspaces.

The null space of matrix `𝐀` is defined as all vectors `x⃗` that satisfy `𝐀x⃗ = 0`, while the Orthogonal Complement of matrix `𝐀` can be calculated as all vectors `y⃗` that satisfy `𝐀ᵀy⃗ = 0`.

The main difference is that to calculate the null space you use the normal matrix `𝐀`, an to calculate the Orthogonal Complement you use the transpose of `𝐀`.
• does khanacademy have anything on math for perspective projection not just orthagonal?
• So to summarize, the projection of any vector in the solution set onto the row space yields the shortest possible solution. Is that correct? What would happen if we were to project the vector onto the null space instead?
• The null space is always parallel to the solution set (i.e. any solution set to a single vector in the column space is always a translation of the null space). What this means is that projecting the solution vector onto the null space would yield the same solution vector
(1 vote)
• Would a line that doesn't go through the origin still be considered a subspace since it wouldn't contain the zero vector? Or should we only care about parametric representations of a line such that they always go through the origin?
(1 vote)
• Not containing the zero vector is definitively what it means to not be a subspace, so a linear that doesn't go through origin does not contain the zero vector, making such a line not a subspace.
• isn't S one of the solution to AX=b? it might not be the shortest solution but isn't it one of the solutions? if it is than S would be memember of C(AT). Soo, this is project of memember C(AT) on C(AT)(r)?
(1 vote)
• Given x and u hat, we can find projection of x by

(x . u hat) u hat = projection of x .

Given projection and u hat, how to we find x vector.

I tried projection matrix, but its not invertible. Please suggest, if any other way?

Thanks
(1 vote)
• It is worth noting that b = [9 18] is a member of c(A) = [3 6]
that is b = 3c(A).
Why am I saying this?
In order for the shortest solution to AX = b to be a member in c(A_transpose), then b has to be a member of c(A).
Sal mentioned this in previous videos. I just think it's necessary to note.

HL.
(1 vote)