Hello, David Friday, I just want to let you know that I am extremely thankful and satisfied with your elaboration on how to write a vector as a linear combination of other vectors, this knowledge will be a great addition to my skill and professional experience which further help me with my studies and professional research again I can not thank you enough for your effort on your lessons, from your best student DR.Данияр
The column of zeros represents the zeros on the right side of the equation. Zero is a number, not a variable. As such, no free variable is needed for this column of zeros.
I'm not familiar with the notation you're using, but here are some possibilities: - If you mean the nullity of T, that's the dimension of the kernel of T. In this case, because of the one free variable and one basis vector, that's 1. - If you mean the nullspace of T, "nullspace" only refers to a matrix. The good news is that the nullspace of the matrix of T, which we call A, is the same as the kernel of T.
Kernel applies to the transformation, nullspace applies to the matrix. The kernel of the transformation, T, is the set of all vectors, x, such that T(x) = 0. The nullspace of the matrix, A, is the set of all vectors, x, such that Ax = 0. Fundamentally, they are the same concept. The difference in terms simply lets you know if you're referring to the transformation or the matrix.
@@davidfriday7498 I'm sure you gave a great explanation but I'm still not sure I understand the distinction. I just think of nullspace as the geometric space made by the span of all the vectors which project into the zero vector after the application of a new basis; a line, plane or whatever. Or unpivoted bases of the matrix after row reduction. Or a zero det. But listen, I self-studied this stuff so I'm not really qualified to comment on the formal stuff.
@@davidmurphy563 I appreciate the backstory of your education. If you don't understand the distinction, don't fret too much. There is a lot of vocabulary-related gatekeeping to higher level math; this is not a battle that needs to be picked. I suppose the thing to keep in mind is that a given matrix can always have a linear transformation, but a given linear transformation doesn't always have a matrix. In the first case, kernel of the transformation and nullspace of the matrix are essentially the same thing. However, in the second instance, because there isn't necessarily a matrix, the term "kernel" would be used without using "nullspace". For example, a derivative is a linear transformation, and the kernel of that transformation is any constant function. You wouldn't be able to effectively model the linear transformation of the derivative as a matrix effectively. Also, to one point you made in your reply: zero determinant is great assuming the matrix is square. However, it doesn't have to be, specifically transforming between vector spaces with different dimensions.
