- #Onto vs one to one examples linear algebra for free
- #Onto vs one to one examples linear algebra how to
So really it's that number x I'd like to find. So we know it's in that one-dimensional subspace, it's some multiple, let me call that multiple x, We know that P, this projection, is some multiple of a, right?
The formula that we want comes out nicely and what's the - what do we know? So if we were doing trigonometry we would do like we would have angles theta and distances that would involve sine theta and cos theta that leads to lousy formulas compared to linear algebra. Let me also say, look - I've drawn a triangle there. That's got to tell us - that's the one fact we know, that's got to tell us where that projection is. That this - the error - this is like how much I'm wrong by - this is the difference between b and P, the whole point is - that's perpendicular to a. The whole point is that this best point, that's the projection, P, of b onto the line, where's orthogonality? What's the - where does orthogonality come into this picture? So where's the point closest to b that's on that line?Īnd let me connect that and - and what's the whole point of my picture now? I'd like to find the point on that line closest to a.Ĭan I just take that problem first and then I'll explain why I want to do it and why I want to project on other subspaces. So just to so you see what the geometry looks like in when I'm in - in just two dimensions, I'd like to find the point along this line so that line through a is a one-dimensional subspace, so I'm starting with one dimension. Let me start by just projecting a vector b down on a vector a.
#Onto vs one to one examples linear algebra how to
I hope this helps clarify how to attack this exercise, if nothing else.OK, guys the - we're almost ready to make this lecture immortal.
Since producing a linearly independent subset is the goal for proving this half of the claim via contrapositive, establishing (1) and (2) will complete this half of the proof. (Unless you're considering a vector space of linear maps or something, like I claim that (1) is linearly independent, and (2) is linearly dependent. Linear independence is a property of a set of vectors, not a property of your map. You have some useful ideas here, but you need to be more careful about the details.Īs an example, in your "=>" direction, you wrote
#Onto vs one to one examples linear algebra for free
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