Nullspace of a matrix

Finding a basis for Nul(A), the nullspace/kernel of A, by row-reducing

[eamon_concannon]

I really appreciate your teaching style. It is such a refreshing change from most math videos!

[CrypticPulsar]

You bring a breath of fresh air to a difficult subject! Thank you!❤

[lightspd]

Dr. Peyam, your teaching style is awesome. Love the energy!

[khbye]

Thank you for the video!!

I just learnt about the nullspace, range and rank-nullity theorem in the more general context of linear transformation functions. Before that, we learnt about the rank-nullity theorem in matrices only...I think I finally got the link to the special case where we look only at matrices
Our notes defined nullspace as N(T)=the set of all the vectors v in vector space V (the domain), such that when you apply T to the v (T refers to a linear transformation, with domain V and codomain W), T(v)=the zero vector in the codomain W

Nullspace of a matrix, is when you let
vector space V (domain) be the set of all column matrices, size n*1 (this is the 'home' of the solution x in Ax=0 !!),
vector space W (codomain) be the set of matrices, size m*1
v be the column matrix x (the solution to Ax=0),
T(v) be the linear transformation function which maps from x-->Ax (the rule of T), where A is a m*n matrix that's left-multiplied onto the x

and the zero vector in codomain W is the zero matrix :)

[jonathanb.burnett]

This was a nice video. Educational and puts a smile on my face.

[eamon_concannon]

On the independence of the solution vectors, I think it becomes clear when we consider that the free variables (y and t here) always appear on their own in the solution vector ( (2y+6t y -2t t) here).
(The pivot variables x and z appear as linear combinations of the free variables in the solution vector). We can write the solution vector as a linear combination of vectors by letting y=0, t=1 and then y=1, t=0 guaranteeing that the vectors are independent.

[holys]

prof there should be a -6 at around 5:43 for the first column 5th component of the column.

[jasonbroadway]

I enjoyed the video, but at 5:43 you might have made an error.

[dhunt]

I'm trying to figure out all the relations of spaces, is the following correct?
space time ⊃ outer space ⊃vector space ⊃ space ⊃ (nasa ∪ esa ∪ jaxa) space exploration ⊃ spacex ⊃ sub space ⊃ personal space ⊃ null space ⊃ monopoly board space
(iI just got the 'go to jail, go directly to jail, do not pass go, do not collect $200' card, darn it!)
Thanks for your videos enjoying both mathematics and silliness!

[ChitranshSagarMTAIE]

Excellent Teaching Style, Superb Sir :)

[nuraisyahbintiomarupm]

that singing hahahhahahha you will always be my basis~

[vijayank]

Dr. Peyam Can you please make a video explaining.. Pointwise and uniform convergence of series, explaining their differences and their relation to continuity with examples.. It would be really helpful.. Because I am having a hard time understanding the idea behind them.

[gvantsasakaruli]

When you started the geometric interpretation i thought somebody was going to draw time!!!

[rolfdoets]

In Holland we also say NUL for 0, thnx for your lecture

[dinhkhoa]

Hi Dr. Peyam, can you prove why the construction of the Null space will always be a linearly independent set? My textbook does not give a proof on this one.

[sugarfrosted]

I always thought it was odd that linear algebra uses different terminology that most of algebra. Null Space rather than the Kernel. I suspect it's because linear algebra developed before abstract algebra.

[StewieGriffin]

The first 2 columns is multiplied by -2

[joluju]

I don't get it. The definition is so simple, I thought we'd just solve for (x, y, z, t) vectors. But then strange things arose, an unknown notation with a vertical bar, undefined terms like pivot, modifying rows with a recipe I never heard of ... Pretty sure the result is the same with a naive method, but where did the method you used come from ? Is there a video somewhere I should see ? Please :)

[joshdilworth]

How do you know by looking at the matrix in RREF that y and t are going to be free variables? is it their lack of a pivot?

[yashj]

What does it mean to be a variable to be free?