Linear Algebra - Lecture 32 - Dimension, Rank, and Nullity

In this video, I define the dimension of a subspace. I also prove the fact that any two bases of a subspace must have the same number of vectors, which guarantees that dimension is well-defined. Finally, I define the rank and nullity of a matrix, and explain the Rank-Nullity Theorem.

[weisanpang]

Hi professor James, all previous videos leading up to this one showed the linear dependence relationship using the x1V1 + x2V2 +...XiVi = 0 equation, where each V term is a vector in the subspace. However, in this video you are showing linear dependence relationship using the "coordinates" of the vectors in the subspace, not the vectors themselves. I don't remember seeing any proof that the linear dependence relationship can be proven using the coordinates instead of the vectors, but here's my reasoning trying to understand it, please let me know if it's correct.
Using vector equation: x1V1 + x2V2 +...XiVi = 0, can be rewritten as:
x1BC1 + X2BC2 + .... +XiBCi = 0, where B is the matrix consists of the basis vectors, C1....Ci are the coordinates.
=> B(x1C1 + x2C2 + + xiCi) = 0
=> Given B is the matrix, then (x1C1 + x2C2 + .... + xiCi) must be = 0

[galben]

Wouldnt the basis for the col(A) be the columns associated with the pivot columns? Now the rows? So it would be [-1 2 3 4], [-2 4 1 2]. Idk man my final is in an hour ._.

[fuadhassan]

I think the row reduction you did is wrong, or let's just say i don't understand why yours is that way

[aditya]

why isn't there a video on change of basis.