Given a linear transformation, find the kernel and range

[@yousiffatohi136]

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[@islamicaestheticvideos]

Man this is the best "how to find range and null space " video for me. THANK YOU SO MUCH.

[@retired5209]

Nice tutoring. I can understand now. Thank you very much

[@user-ft8io7dz2c]

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[@user-ft8io7dz2c]

true

[@user-ft8io7dz2c]

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[@raghavkumarsingh4222]

Iam confused in finding Range of T if T:R²-->R³...plz help

[@davidfriday7498]

Respectfully, if you don't give me the definition of the transformation, there is literally nothing I can do to help.

[@overclocked7260]

awesome video

[@veronicanoordzee6440]

Your math is okay, but why don't you give a short description of the objects you are describing? Teaching is not for KZbin-amateurs.

[@MrVitoCorleone]

Great

[@davidmurphy563]

Is this an American thing to say kernel instead of nullspace? The latter is a much better term in my humble.

[@davidfriday7498]

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.

[@davidmurphy563]

@@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.

[@davidfriday7498]

@@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.

[@captainnobody4960]

Your missing a free variable for the column with all zeros

[@davidfriday7498]

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.

[@arkojyotidutta5890]

​@@davidfriday7498❤

[@matarmqds307]

How i can find null (T)?

[@davidfriday7498]

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.