Image and Kernel

Now that we've learned about linear transformations, we can combine this with what we know about vector spaces to learn about the concepts of image and kernel. Let's get a closer look!
Script by Howard Whittle
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[@arfinsyed]

The most simple explanation that I have found on the entire Internet that explains Image and Kernel after searching for 2 weeks!! THANK A MILLION!!!!!!

[@phantorang]

Your haircut looks great, if you were wondering. Other than that, I got no idea what this is about, I guess I will have to read up on it 😁 👍✌

[@a.human.]

Definitely recommending this video to all my friends and future students studying Linear Algebra

[@ion9328]

Finally someone show what it actually means so you can understand what you are actually searching for, thank you!

[@ahmadyogi1340]

Wow this is the best explanation of kernel and image on youtube, thank you so much Professor Dave!

[@kmishy]

Prof Dave, You are sweat heart of every students who loves mathematics.

[@sohamsantra7548]

Professor Dave = Guaranteed success in academics!

[@hcos8139]

Thank you so so much. Every time I have an exam I read the 70 pages with of chapters and I barely understand anything. 5 minutes of your video explains it PERFECTLY clear. Thank you so much

[@lukevillanueva1494]

My professor went through the definitions of kernel and image way too quickly in his not-recorded lectures. This saved me, thank you

[@__-ni1kz]

Getting halfway through linear algebra and having the prof say "for those of you in engineering, this everything past this point isn't useful to you" really took the wind out of my sails. I hate having to learn these complex abstractions for no reason, when I could be spending my time studying for finals that have utilization in my field. Thank you for a straight forward explanation, once we exited euclidean vector spaces the math became harder to conceptualize.

[@RyantheCanuckpirate]

This is the first day in my life that I understood exactly what the "kernel" was. Even managed to do the practice question. Baby steps! (By the way, I love the inclusion of the knowledge confirmation. Too many times have I watched a math video and thought I got it, only to blank out on an exam. Having a warm-up question lets me know whether or not I need to learn the base concepts more, or continue doing questions)

[@JoseAlvarez-dl3hm]

Totally awesome, I finally get it, thank you very much.

[@danielmoss7133]

Fantastic video! Far better explained than in my university lectures

[@BO55]

Wow, what an explanation. Even after listening to my prof's video multiple times, didn't get this understanding. This video explained my doubts about why are trying to equate the L(s) = 0 to find the kernel in a pretty elegant way.

[@KidNamedVashin]

Thank you, Great explanation! Coincidentally I have an exam about this next week

[@ysabellarivera8166]

lmao, i have a final ToMORROW

[@aurumstinger4479]

Mine is on Friday

[@vaishnavisharma9467]

Love your videos. Nice haircut btw. Looks great. Can you please make a few videos on calculus too

[@ProfessorDaveExplains]

I made a 35 part calculus series, check it out!

[@vaishnavisharma9467]

@@ProfessorDaveExplains sure thank you

[@syunegevorgyan]

O thank you. You can't even imagine how you helped me. I was about giving up when I saw your video and I started to understand this theme. thanks very much again

[@geoffrygifari4179]

your explanation has been very concise.... awesome

[@Demon0228]

This helped immensely and simplified the concept so nicely. Appreciate it

[@ranitchatterjee5552]

People like you deserve great honor!!!

[@wecanplay5363]

should'nt kernel be 0,0,0 for the last question as the transformation is from 2 to 3

[@2K_Curse]

By definition , set of vectors in R2 that give 0 vector in R3 will be the kernel. That is why because these vectors are from R2.

[@wecanplay5363]

​@@2K_Curseokie, 😃👍

[@hamzamehmood4099]

Very nice teacher

[@kimkakyeong]

This guy just turned my last week's 1hr long linear algebra lecture into a 5 min video... Thank you!

[@Harminder_Saini]

Same lol

[@just4simplegg428]

Thank you, professor, you are truly a Mater. You can explain tough topics in a very easy way! Happy Holidays

[@Firelord375]

For Dave and all the people that make these videos happen, thank you so much!

[@DheaFikyFatchaturRizky]

thankyou for this amazing

[@ts.nathan7786]

If you give graphical representation of the transaction L : R^3 --> R^2, and pointing out the kernel, it would be useful.

[@abhijitdeokule1986]

best video on image and kernel

[@CSBPRAKASHDEEPAK]

Thanks Professor Dave.

[@anky137339]

Hi All , What is the intuition for having Kernel mapped to space W is 0 ? is this shows 0 loss of information when transformed ? AND this is great video explaining concepts in 5 mins

[@lightok111]

Really well explained Sir. I'm grateful

[@jademath910]

If we have two linear operators on n-dimensional vectors space, say f and g what would be the image(f+g), image f , and image g? can we say that image f = image g?

[@TheSmartgutt]

What is the difference of the Null space and the kernel then?

[@mehdimansouri1547]

simple and amazing

[@ranbirsingh494]

nice man great

[@sgtivo7330]

This gets super funny if you watch it at 1.5x the speed... given you already understand the Concept and your head doesn't start smoking.

[@animeshsinghal3405]

Elegantly explained. Thanks a lot

[@aishwaryabimaljoy6642]

if the vectors of the form like wise is not given in the question how to find the image

[@catalinaionascu171]

Thank you

[@mladizivko]

Amazing, right to the point. Thanks!

[@khamishoufar1019]

in machine learning, we have a concept of the kernel in the context of similarity measure between two observations! what is the intuitive explanation about that ..Thanks

[@curtpiazza1688]

Thanx Prof Dave!

[@put.7070]

Thanks!

[@nomaetamamo6479]

nice video, thanks alot

[@nghiatrinh8753]

this was very helpful

[@DogeCharger]

Quick question: If I understand kernels correctly, then that means that the Dimension of them will generally be 1? And the Dimension of an Image will generally be what it is equal to -- so if it equals ax^2+bx for example, then the dimension will be 2?

[@johnmccrae2932]

It turns out that the dimensions of the image of a linear map and the kernel of a linear map depend on a property of maps called "rank". Pretty much, if you think of a map as a matrix multiplying a vector in the domain, the rank is how many of the matrix's columns are linearly independent (not multiples of each other). The dimension of the image is the rank, and the dimension of the kernel is the number of columns in the matrix minus the rank. Definitely check out Gilbert Strang's linear algebra lectures on MIT OCW if you want to know more (I realise my explanation might be quite dense).

[@Ahmedibrahim0100]

Thanks alot

[@bartman999]

Why is the rest of this course made private?

[@ProfessorDaveExplains]

I release one per week!

[@bartman999]

@@ProfessorDaveExplains Good to know! Looking forward to the rest.

[@tadabae]

i finally understand the kernel after soooo long

[@sajithpriyankaragunawardan4530]

Too sweet example love much

[@nabanitagoswami6652]

Sir pls can you teach s..what is rank of matrix and how to find it

[@mohammadidreesbhat1109]

Great...

[@salamlababidi3042]

GOAT

[@bella-rp2rw]

You are the English version of Elia Bombardelli for italian students

[@the_gis_madiam]

professor dave has saved another poor student's braincells! (me)

[@bimsara3824]

easily undersood whole lecture within 5 minutues

[@leekimjinyoon4473]

What’s happening?

[@jamesevans2211]

Second!

[@3Space1time]

First view First comment First like

[@gunjansharma6603]

don't tell me i skipped one test and tried and gave up on this topic three times only to learn it in 5 minutes whaaat

[@ronycb7168]

always at .8 or .75 X prof just commented here attendance and for the posterity of mathkind. bad word play sorry but in a hurry to finish this stuff already too late...

[@bencnu9763]

sooooo, where does popcorn come from......i tihnk i missed a step

[@TheAllen501]

thx, my prof is useless

[@AyushKumar-cq2ze]

😘😘😘

[@nihalkpdb]

No

[@KG16888]

not well explained, still confused

[@ProfessorDaveExplains]

watch again