Support this course by joining Wrath of Math to access exclusive and early linear algebra videos, plus lecture notes at the premium tier! ruclips.net/channel/UCyEKvaxi8mt9FMc62MHcliwjoin Linear Algebra course: ruclips.net/p/PLztBpqftvzxWT5z53AxSqkSaWDhAeToDG Linear Algebra exercises: ruclips.net/p/PLztBpqftvzxVmiiFW7KtPwBpnHNkTVeJc
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.
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.
Support this course by joining Wrath of Math to access exclusive and early linear algebra videos, plus lecture notes at the premium tier! ruclips.net/channel/UCyEKvaxi8mt9FMc62MHcliwjoin
Linear Algebra course: ruclips.net/p/PLztBpqftvzxWT5z53AxSqkSaWDhAeToDG
Linear Algebra exercises: ruclips.net/p/PLztBpqftvzxVmiiFW7KtPwBpnHNkTVeJc
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!
I love Strang's take on this, especially summarised like this. This is a Godsend, thank you!
very ituitive explanation ever seen so far. Really good materials for newbies in LA. Recommended!
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.
great vid God bless your work
got a 91% on my exam, thankyou....
This guy… this guy teaches
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.
My professor wishes she was as good as you
12:21 left in a outtake
Thank you, just used RUclips's built in editor to cut it out, hopefully once it finishes processing it will be fairly seamless.
this shit makes no sense im gonna fail my LA test tomorrow
well, good luck! Let me know if you have any questions!