Linear Algebra Example Problems - Matrix Null Space Basis and Dimension

[@danielvanbibber991]

This is the clearest explanation on this subject I have seen thus far! Thank you!

[@AdamPanagos]

Thanks for the kind words, I’m glad you enjoyed the video! Make sure to check out my website adampanagos.org for additional content (540+ videos) you might find helpful. Thanks much, Adam

[@cosmicangel233]

Yes. This is literally exactly what I needed.

[@AdamPanagos]

Glad I could help, thanks for watching!

[@Thefakisgoinon]

9 minutes of this video is better than 4hours worth of lectures in university

[@AdamPanagos]

Thanks for the kind words, I’m glad you enjoyed the video! Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks much, Adam

[@password6975]

These videos are some of the best linear algebra content avialable on the internet

[@AdamPanagos]

Thank you for the kind words!

[@gauravbadhani159]

Great...after 2 hrs i have exam...this helped a lot easy explanation...thanks a lot...

[@AdamPanagos]

Glad I could help; hope the exam went well! Make sure to check out my website adampanagos.org where I have a lot of other resources available that you might find helpful. Thanks, Adam

[@mariettarandolph3122]

Sorry, at timestamp 4:43. Multiplying A by = . Correct? Forgot to mention that your Linear Algebra videos are by far the best!!

[@AdamPanagos]

Ah yes, great catch! I guess I got a little carried away with the zeros there. =) Definitely should only be 4, not 5. Thanks for noting and glad you like the videos. Adam

[@atharvaupasani247]

Cleared every doubt .

[@AdamPanagos]

Glad I could help, thanks for watching. Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks, Adam

[@backtonature1923]

thanks a lot sir, you are better than my teacher at my university.

[@AdamPanagos]

Thanks for the kind words, glad I was able to help! Best, Adam

[@Zuwwar]

You are a blessing and a great teacher

[@AdamPanagos]

Thanks!

[@SpitfireAce]

Perfectly explained! Exactly what I needed.

[@nickyeung1641]

Just love your teaching videos, Adam!!! I’m a math teacher too, I’m wondering what software / app you’re using in your explanation? It looks so smooth during your teaching!! May I know it ? Thank you very much . 🙏

[@AdamPanagos]

For most of my videos, I use Doceri (an app for my iPad, apps.apple.com/us/app/doceri-remote/id412443803?ls=1) that lets me pre-record all the writing. Then, I can "play back" the writing while I record the audio. It’s a very useful tool for me. Hope that helps, Adam

[@nickyeung1641]

@@AdamPanagos thank you so much, Adam!!!👍👍

[@raveendrasothykeerththiga9231]

great explanation sir...

[@AdamPanagos]

Thanks for the kind words, I’m glad you enjoyed the video! Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks much, Adam

[@MegaDurocher]

I appreciate you filling in the missing steps from my math notes

[@AdamPanagos]

Glad I could help, thanks for watching. Make sure to check out my website adampanagos.org for additional content you might find helpful. Thanks, Adam

[@phuclehoang8024]

thank you a lot sir. You make it easy for student to understand.

[@AdamPanagos]

Glad I could help, thanks for watching!

[@DarknessIsThePath]

I learned what nullspace and dimension is in 9 min than I did in 1 week lol, excellent video!

[@AdamPanagos]

Awesome! Glad I could help. Thanks!

[@prospermukura3594]

Great teacher indeed...this video eradicated my confusion on this topic...thank you

[@AdamPanagos]

Glad I could help, thanks for watching. Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks, Adam

[@baphomet705]

Thank you so much. It was of great help. Could you please recommend some reference books on this topic?

[@AdamPanagos]

My favorite linear algebra text for undergraduates at the moment is www.amazon.com/Linear-Algebra-Its-Applications-5th/dp/032198238X. Hope that helps. Adam

[@suryat1103]

Excellent Explanation. Keep up good work. Thanks Surya

[@AdamPanagos]

Thanks for the kind words! If you found the video useful make sure to check out my website adampanagos.org where I have a lot of other resources available and consider supporting development of new videos through www.patreon.com/agpanagos. Thanks! Adam

[@Santiagodespa]

amazing teacher really helped thanks

[@AdamPanagos]

Glad I could help, thanks for watching! Check out my website adampanagos.org where I have about 400+ videos you might find helpful.

[@ashrule2034]

great video, very clear explanation. thanks!

[@AdamPanagos]

You're very welcome, thanks for watching. Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks, Adam.

[@salmaabobakr1406]

you are great, thank you

[@deepakpandey9935]

thanks a lot!! great explanation

[@AdamPanagos]

You're very welcome, thanks for watching. Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks, Adam.

[@marufbillah4857]

You are a blessing and a great teacher. tnx a lot

[@AdamPanagos]

Thanks!

[@sreejaramagiri7630]

Thank you so much

[@sreejaramagiri7630]

It's help me a lot

[@AdamPanagos]

You're welcome, glad I could help.

[@achrafelkassih2930]

thank u so much

[@AdamPanagos]

You're very welcome, thanks for watching. Make sure to check out my website adampanagos.org for additional content (395+ videos) you might find helpful. Thanks, Adam.

[@fenerbahcesk5712]

if there is no free variable so if all row has pivot what will we do what is the dimension of nullspace

[@cathylin1188]

Do we need to check whether these 3 vectors are linearly independent?

[@AnhNguyen-sl7nx]

Are the vectors we have at the end (in this case, 3 vectors) always linearly independent? They have to be if they make the basis of the null space.

[@AdamPanagos]

Yes, using this approach they will be.

[@Wexico8]

Thank you so much!!! 💚

[@AdamPanagos]

You’re welcome, thanks for watching. Make sure to check out my website adampanagos.org where I have additional resources you might find useful, and consider supporting development of new videos through www.patreon.com/agpanagos. Thanks! Adam

[@josephgrasso5785]

I thought the vectors must also be linearly independent to form a basis?

[@AdamPanagos]

Either definition is used. Some texts/resources require independence of the vectors, others just require spanning. Hope that helps. Adam

[@gideonbuckwalter4128]

Great video! Thanks for posting. (At 4:43, shouldn't the zero vector have only 4 elements?)

[@Therysin]

nope, it has the same number of elements as the number of columns

[@jerrerock]

@@Therysin why though a 4x5 matrix X 5x1 = 4x1 no?

[@Therysin]

@@jerrerock idk man, it's been two years. I got my engineering degree now lol

[@purandarparitosh]

brilliant !!!! :)

[@AdamPanagos]

Thanks and thanks for watching. Best, Adam

[@damianrauch5460]

What if you can't write your basic variables with your free variables. Meaning I have a basic variable with other free variables?

[@AdamPanagos]

You can always write your basic variables as a function of your free variables.

[@kjitwasntme3]

how do you have 3 vectors with only a geometric multiplicity of 2? I thought the number of vectors you need to have needs to equal the nullity of the reduced matrix

[@AdamPanagos]

I'm not sure if I follow how "geometric multiplicity" has anything to do with this. As shown in the video, the original matrix A is row equivalent to matrix B which has just 2 pivots. Since there are 5 columns, then there are 5 - 2 = 3 free variables which correspond to the 3 vectors that form the basis for the null space; e.g. x1 and x2 can be written in terms of x3, x4, and x5.

[@TK-wp3iw]

Can anyone please explain 8:30 to 8:40 ??? Please

[@AdamPanagos]

The dimension of any space is always equal to the number of basis vectors it takes to represent the space. In this case, we have 3 basis vectors that were found. So, the dimension of the null space is 3. In general, if you need N basis vectors, you have an N-dimensional space. Hope that helps, Adam

[@TK-wp3iw]

@@AdamPanagos okay!! Thanks

[@shayantnandi7010]

At time stamp 06:53, X3 for the 3rd row and 1st column should be 1 not 0..??

[@AdamPanagos]

Yes, you're correct at that time instant. Note that a few seconds later I caught the mistake while recording and got things corrected. Sorry for the mistake. Good catch!

[@mariettarandolph3122]

I think your zero matrix should have only 4 zeroes instead of 5.

[@AdamPanagos]

What zero matrix? Can you tell me what time in the video you're referring to?

[@kharerishit]

thank you for this

[@AdamPanagos]

You're very welcome, thanks for watching. Make sure to check out my website adampanagos.org for additional content (600+ videos) you might find helpful. Thanks, Adam.