Can a linear transformation from 2D to 3D be onto?

Can a linear transformation from R2 to R3 be onto?

Check out the follow up video for the solution!

Learn about topics in linear algebra like image, linear transformations, surjectivity, and more!

Made by
Nic Swanson

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[quanjano]

My guess is no, because even if you have 3 dimensional basis vectors, you only have 2 of them, which can’t span R^3

[minimath]

Linear algebra is beautiful when animated!

[suomeaboo]

Today I learned "onto" can be an adjective in math.

[elvissalcinovic]

I love this… I hope to one day learn Manim and provide exceptional educational content like you and 3blue1brown

[vnever]

There should be more videos like this.

[camcorl]

No because we have a theorem that says for T:V->W a linear transformation Dim(V)=Dim(Im(T))+Dim(Ker(T)) and here Dim(V) is 2 and Dim(Im(T)) is 3. 2=3+a positive value since Dim is always superior or equal to 0.

[MohammadHBakr]

Well based on what I know in linear Algebra a transformation from a higher dimension to lower dimension can be onto if it's simply a projection, the other away around doesn't work tho, as a transformation from 2D to 3D can only result in either a line or a plane, so yeah
From 2D to 3D can't be onto.

[SniffySnoffy]

You need a space-filling curve to map onto.

[utube_int]

It's goood. Are you planning to continue?

[cowgomoo]

so im guessing NO because you need 3 basis vectors to fill up 3d space and we only have two basis vectors that T can work with. a third one isnt going appear no matter what T is.

nice video. i hope i understood it correctly and came up with the correct solution.

[Abhyaskul]

can you please tell using which software or site you made this video?
like visual graph part and rest of videos, is it in microsoft powerpoint?