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# Another example of a projection matrix

Figuring out the transformation matrix for a projection onto a subspace by figuring out the matrix for the projection onto the subspace's orthogonal complement first. Created by Sal Khan.

## Want to join the conversation?

• Just a bit curious if this is a special case. At we have seen that V is a null space of [1 1 1]. What would be the situation if V is not a null space or the equation is something like x1+x2+x3=5.

Does it mean that cannot use the Proj v(perp) x for simplicity. Does it mean we would need to persist with the method being described in the previous video? • What is an orthoplement?? I got this on a test! • At , Sal said, "And we could actually just take that out. It's a 1 by 1 matrix, which is essentially equivalent to a scalar". But isn't this sloppy notation, as 1x1 matrices aren't the same as scalars? Is this considered bad practice? And if I were to directly "take [scalars] out" of 1x1 matrices, would it always work or are there situations where I need to watch out? • A 1 * 1 matrix's purpose is the same as a scalar, i.e. representing a single number. In reality, the operations for a 1 * 1 matrix are so limited that I've never actually seen a "1 * 1 matrix" in my textbook. It's this scalar-but-not-a-scalar, matrix-but-not-a-matrix sort of thing that can be only be added with other 1 * 1's, and can only be multiplied by some matrices (which kind of defeats the importance of scalar multiplication in linear algebra).
• At Khan said the orthogonal complement to V is a line. I dont quite understand since I think other lines parallel to span([1,1,1]) can be the orthogonal complement to V, too. Can someone please explain? • Is orthonormal the same as orthogonal complement?
(1 vote) • an orthonormal set is a set of (linearly independent) vectors that are orthogonal to every other vector in the set, and all have length 1 as defined by the inner product. an orthogonal complement is done on a set in an inner product space, and is the set of all vectors that are orthogonal to the original set and is in the inner product space. notice a regular vector space has no definition of orthogonal.
• At Sal says that the vector v is by definition the projection of x on to v. How is that so? What definition is he referring to?
(1 vote) • at , shouldn't the left side be [Dtranspose] x [D]?
(1 vote) • Hi,

I am making a 3D program, and here is what I am trying to do;
1. Since we are looking at the 3D program through a 2D screen, I want to "project" the coordinates of the matrices onto the 2D plane (screen). This will give my program perspective I believe.
2. So to do that I need to find a subspace that is the plane centered at z = 0 (where x & y are free variables), and then find it's basis so I can plug it into the equation to find the projection.
3. But, I'm stumped for some reason. I can't seem to do this. Any help?

Summary; I need to find the basis for the plane centered at (z = 0).

Here is what I don't understand;
Should I have the basis be in R2 or R3?
` [1, 0]A = [0, 1]`

` [1, 0, 0]A = [0, 1, 0] [0, 0, 0]`   