Linear Algebra 34 | Range and Kernel of a Matrix

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This is my video series about Linear Algebra. I hope that it will help everyone who wants to learn about it.
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(This explanation fits to lectures for students in their first year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[@MichaelWilliams-ow9ue]

I was literally thinking about kernels right when I saw that this video had just been uploaded four minutes ago.

[@brightsideofmaths]

Cool :)

[@MrWater2]

Sorry I didn't get what to prove where ran(A)=span(a1, ..., an). Is it not just a definition? Because the range define as Ax is the linear combination of the columns of A with x in the domain (Rn). And the span it's just the set of vector u in Rn (if M is in Rn) such that there are lambdas_j in R and u_j in M where every u can be write as a linear combinations of another vectors u_j. So the u_j are the a_1, ..., a_n and the lambdas are the x_i, ..., x_n so every vector u can be write as linear combination of the vectors of A. Am I correct?
Mmm! Also if A is not Linear independent suppose Rk with k

[@brightsideofmaths]

Ker(A) and Ran(A) are linear subspaces :)

[@user-eo5gv9zn8m]

What is name of your whiteboard you use ?

[@brightsideofmaths]

There is a video of mine where I explain it: tbsom.de/s/up