Procedure to Find a Basis for a Set of Vectors

Procedure to Find a Basis for a Set of Vectors. In this video, I start with a set of vectors in R_3 and find a basis for those vectors. The basis is NOT necessarily unique!

Category:

Education

Tags:

• linear
• algebra
• basis
• vector
• independent
• independence
• span

[kevinp8001]

Please another example where the result be 3 columns in diferente postion, no just the column 1, 2 ,3. thanks.

[patrickJMT]

@kevinp8001 just rearrange them and knock yourself out : )

[autogordel]

Thankgod for people like patrickJMT, making things understandable that teachers seem to make much harder on purpose.

[LAnonHubbard]

Thanks. This has been on my watch list for a while and finally got round to it. Always good to see more examples on this.

[mesmeriz3]

If only i found this video earlier, im sure my concept on linear algebra would have been much clearer!

[skdbstjddlek]

Also, I understood the dimension of a vector space as the number of vectors in the basis. So if a basis be {v1,v2, ... ,vn} for a typical vector space, then its dimension would be n. In this case, if I get total 3 basis for each U and W with one of them the same, then the size of the basis for U + W would be 5. ({v1,v2,v3,v4,v5}) So the dimension of U + W would be 5? or does it not matter since the spanning sets are all 4 dimensioned.

[skdbstjddlek]

However when I checked the answers, they have done it by setting x4 = 0.

Which results in total 2 solutions ( 1 from U intesection W and the other from U only)

Although the answer notes "many other possible answers", I wonder if the solution I found is valid.

[skdbstjddlek]

So yeah, for part (i) you do the row operation for all the vectors U , W and get a general solution. (say its v1)

But its (ii) that im confused.

What i did was find the general solution for U itself, say the solution are v2,v3.

Then the basis for U would be {v1,v2,v3}.

How I found the solution for U is, after the row operation, I set x3 =a and x4 = b which gives two solutions, and 3 in total including U intersection W.

Similar for finding the basis of W.

[skdbstjddlek]

Im studying basis for myself, but having a hard time on one question.

The question is:

Let U and W be the following subsets of R4( 4th dimension of real numbers)

U = {x1+x2+x4 = -x1+x2+x3 = 0},

W = {2x1+x3-x4 = -x1+2x2+x3+x4 = 0}

(i) Find the basis of U intersection W.

(ii) Find the basis of U and a basis of W, both containing your basis of U intersection w.

(iii) Find the basis of U + W containing your basis of U intersection W.

[JoshPity]

My algebra professor called that gausjordan or something crazy, where were you when I was doing this last year! At least I found you for my applied calc class haha! Keep on going on.