Finding the matrix of a linear transformation and figuring out if it's one-to-one and/or onto. Check out my linear equations playlist: • Linear Equations
For onto, I think about the space of all inputs (x) as an island and the space of all outputs as another island. I imagine the transformation A to take anyone in (x) to a place in the outputs (b). And onto means that there will always be an inverse bridge that takes anyone in (b) to a place in (x).
Yay! Great video as always! The answer to those 2 questions is very dependent on the space considered. If your image space is {(0,x,y,z) l (x,y,z) ∈ lR^3}, then the results differ, don't they?
Hello! Thank you so much for your video, that was really helpful! But I have one question when mentioning the pivot positions in the rows and columns, do we consider only the coefficient matrix or the augmented matrix?
Why do English speakers always use "1 to 1" and "onto"? Here we always say injective and surjective (and bijective instead of "1 to 1 and onto" for that matter). Just something I always wondered.
Beyond linear algebra it's pretty much never used. Though I can defend 1 to 1 because it's often useful to talk of " to ". Though the phrase "maps onto" instead of "maps surjectively" is still used.
I remember the theorem. For a linear map from a vector to itself: injective if and only if surjective. It's a consequence of the dimension. Of course, that's not really good to bring up yet, pedagogically.
Hi Dr. Peyam! I haven't seen all your linear algebra videos; can you tell me if you already made a proof for the Rouché-Capelli/Kronecker-Capelli/Rouché-Fontené/Rouché-Fröbenius theorem? (All the names are different ways for naming the same theorem). If not, can you make a proof? I really need it to understand my classes. Thank you very much if you read this.
@@drpeyam It's funny because in Spain, where I live, is like a fundamental theorem, however, I always prefer to search math content in English but I barely see videos about that theorem. I would be so pleased if you dedicate one video for it :)
@@drpeyam It is a great theorem for discussing linear systems of equations with some parameters. Let A be the coefficient matrix and A* be the augmented matrix of a system of linear equations; let n be the number of variables the system has. Then the theorem states that if rank(A)=rank(A*)=n, the system has exactly one solution; if rank(A)=rank(A*)
Would this be a valid and simpler argument without considering the associated matrix A at all? Every output of T has 0 in the first coordinate so it's clearly not onto. Furthermore, it's range is at most 3 dimensions so it must be many to one.
That's the terminology I learned in first course Calculus and Linear Algebra and the one I mainly use. One-to-One is also ok, but 'onto' hasn't even got a proper translation in my daily language. However as long as English is concerned, I think all these words are equally worthy. Probably 1-1 and onto are even more suitable for a first course
