Linear Algebra Example Problems - Vector Space Basis Example #1

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In this video we determine if a set of given vectors is a basis for R3.

We know that in general, a basis for Rn requires n linearly independent vectors. Since we're given 3 vectors in this problem, we require these 3 vectors to be linearly independent if they are to form a basis for R3.

Two different methods are used to check for linear independence of the vectors. In the first, we construct a matrix and perform row operations to show that we obtain a pivot in each column. This implies that the only solution to Ax = 0 is the trivial solution (i.e. x = 0) and thus the vectors are independent.

In the second method we compute the determinant of the matrix. Since the determinant is non-zero, the vectors are independent.

Since we've shown that the three vectors are linearly independent, then they form a basis for R3.

[MichaelUtah]

im not sure if im the only one that saw this, the row second row operation to get the third matrix is not correct. E3=E3-6E2, the last element in the third row should give -17

[ijazkhanniazi]

You are the best math teacher for me on planet right now . Thanks

[minhgiangoan]

I don't know what I have to say to you. Your tutorial have been helpful so much. If you had a tutorial about Algorithms, I would watched that. Thanks so much, one again.

[johnmosugu]

Wow! Thanks, Adams for making this look so simple. This video nailed it for me. Well appreciated.

[noel]

Brilliantly described, better then university lecturers, thank you

[vibudhdixit]

Sir, all your videos have been really really helpful for me! just wanted you to know that you are doing a great job :').

[BoZhaoengineering]

I like the linear algebra video very much. Because it interpreted the idea behind the computation.

[anindyamondal]

Your videos are very helpful for end Semester Mathematics Examinations in Engineering. We will be much helped if you prepare some solved practice sets for us in future.

[neelakshigoel]

Thankyou for the alternate method of determinants

[codingwithelhacen]

Alway planning to watch your great videos. Thank you

[desorae]

Thank you so much! I’ll surely be passing my test

[szymuswitkowski]

Why does the first method work? Why is it that if after doing these operations the resulting vectors are linearly independent then the originals are as well?

[katyperryaf]

if A is a matrix and det(A)=/=0 are the columns always linearly independent? or are there conditions that need to be satisfied, for instance, A has to be an nxn matrix.

[charlesamofordjuoh]

May you please post a couple of more examples and examples on finding if the vectors are dependent.

[xXBRDNXx]

Don't you need to prove the set spans R3 as well?

[eddieed]

Correct me if im wrong but is it true that the letter " R" stands for real number?

[Garbogebi]

What program do you use to write this?