Basis and Dimension 

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When you work with column vectors and perform row operations on matrices, the operations mix the components within each vector. For example, when you add or subtract rows representing column vectors (like [x, y, z, w]), the operations combine elements from the same vector. This can lead to combinations like x + y within the same vector, which doesn't maintain the original components separately. However, when you use row vectors in row operations, these operations don't mix components of the same vector. Each row represents a different vector, and the operations performed on rows don't blend or combine elements within a single vector. you just add or subtract or exchange vectors, and it is fine. So , that's why you cannot use the final columns when you eliminate using column vectors. You do not perform linear operations between different vectors, rather you mix a vector with itself

Thanks a lot for showing the case of these vectors as columns. I had solved the matrix for pivots and chosen first three columns of the Echelon matrix as the basis. But clearly, as you pointed it out I was wrong. Awesome tutorial.

I just fell in love with this teacher ,you gave me such a great understanding

I think this girl and the Asian one are the best TA so far in any MIT ocw

Thank you Ana and MIT!

thanks for pointing out about using the transpose matrix to solve the problem. that was exactly my question

Thanks MIT for sharing such a great teacher and his teaching with the Whole World..A Learner From India..

thanks a lot for the clarification at the end

It would be good to know why you can use the rows of the echelon matrix when doing vectors-as-rows, but can't use the columns of the echelon matrix when doing vectors-as-columns. The fact is stated, and justification is given in terms of the example ("not enough numbers"). But the reason why the methods aren't symmetrical is not explained. I believe there ought to be a good geometric explanation for this, or at least something in terms of the definitions of the spaces.

as there are 5 column vectors and each vector belong to R^4(we can have almost 4 linearly independent vectors for R^4) so don't even need to check if they are dependent by doing gaussian elemination.

excellent way of teaching👏👏👏👏

Very clear. Thank you very much!

love this course

please confirm (1,1,-2,0,-1) row vector or column vector, while soling TA taken as Row Vector is it correct

Dont We take columns in basis?

Can I solve it by finding the rref of the given matrix

So if I wrote the vectors as rows and did the elimination, I can directly use the final 3 rows (with pivots)?

great recite!

thank you so much

Thank you!

Thankyou very much

Why is that the elimination of column vectors changes the column space ( 7:06 ) but the elimination of row vectors doesn't change the space (4:40) ?

Why do those vectors become rows instead of columns of the matrix? Is it because they have to follow the rule of forming an mxn matrix where m < n? I'm confused.

Thank you :)

Hey, I know this is a stupid question. What is the transpose of this universe?

Vow....

Thank you!