The Four Fundamental Subspaces and the Fundamental Theorem | Linear Algebra

[@WrathofMath]

Support this course by joining Wrath of Math to access exclusive and early linear algebra videos, plus lecture notes at the premium tier! kzbin.info/door/yEKvaxi8mt9FMc62MHcliwjoin Linear Algebra course: kzbin.info/aero/PLztBpqftvzxWT5z53AxSqkSaWDhAeToDG Linear Algebra exercises: kzbin.info/aero/PLztBpqftvzxVmiiFW7KtPwBpnHNkTVeJc

[@胡家瑋-w7n]

This guy tried to teach ALL the content of our whole semester's Linear Algebra in 20mins and he done it better than my prof!

[@jamesfehrmann]

Awesome video. There's a lot going on in a sentence like "the solutions to Ax=0 are the vectors in R^n that are orthogonal to every row vector of A." And it might not be obvious why the dot product is an effective demonstration of this. It might help build some intuition for this by remembering that in 2D/3D space the dot product of two vectors shows the projection or shadow one vector casts on the other. Orthogonal vectors can't cast shadows on each other because they are perpendicular. Thus their dot product is zero and we can use Ax=0 to to find these vectors. These vectors are simultaneously the definition of the null space and orthogonal to the row vectors. Which is why we can say the null space and row space are orthogonal, and why the dot product is tool to get there.

[@SeeTv.]

15:45 No, the row vectors are not necessarily a BASIS of row(A) (unless r=m), but they are a SPANNING SET of row(A) and this is the only thing we need for the proof.

[@yizhu5275]

very ituitive explanation ever seen so far. Really good materials for newbies in LA. Recommended!

[@yojacq]

great vid God bless your work

[@nathanisbored]

12:21 left in a outtake

[@WrathofMath]

Thank you, just used KZbin's built in editor to cut it out, hopefully once it finishes processing it will be fairly seamless.

[@JoshuaRoose]

this shit makes no sense im gonna fail my LA test tomorrow

[@WrathofMath]

well, good luck! Let me know if you have any questions!