We colored the quarters of the square in different colors to help visualize how points within the square were mapped. The determinant is 4 even though it seemed it was streching everything by a factor of 2.
And the determinant was positive even though it rotated everything so that points on the right are mapped to points on left and points on the top are mapped to points on the bottom. Shouldn't we have gotten a negative determinant? This quadrupling of the area is reflected by a determinant with magnitude 4. If we go counterclockwise around the perimeter of the mapped square, we still encounter the colors in the order red, green, yellow, blue.
As shown below, it maps the square into a parallelogram. Moving counterclockwise around the perimeter of the parallelogram leads to the opposite color order red, blue, yelow, green. There is no way to stretch and move the original unit square into the parallelogram without taking it out of the plane and flipping it or somehow moving the region through itself. It's not necessarily all of the things that we're mapping to. For example, the image of Rn under transformation, maybe it's all of Rm or maybe it's some subset of Rn.
The way you can think about it, and I touched on this in that first video, is-- and they'll never, or at least the linear algebra books I looked at, they didn't specify this-- but you can kind of view this as the range of T.
These are the actual members of Rm that T maps to. That if you take the image of Rn under T, you are actually finding-- let's say that Rm looks like that. Obviously it will go in every direction. And let's say that when you take-- let me draw Rn right here. And we know that T is a mapping from Rn to Rm. But let's say when you take every element of Rn and you map them into Rm, let's say you get some subset of Rm, let's say you get something that looks like this. So let me see if I can draw this nicely.
So you literally map every point here, and it goes to one of these guys. Or one of these guys can be represented as a mapping from one of these members right here. So if you map all of them you get this subset right here. This subset is, this is T the image of Rn, the image of Rn under T. And in the terminology that you don't normally see in linear algebra a lot, you can also kind of consider it its range. The range of T. Now, this has a special name. This is called -- and I don't want you to get confused -- this is called the image of T.
Image of T. This might be a little confusing, image of T. So this is sometimes written as just im of T. Now you are a little confused here, you are like, before when we were talking about subsets, we would call this the image of R subset under T.
And that is the correct terminology when you're dealing with a subset. But when you take, all of a sudden, the entire n dimensional space, and you're finding that image, we call that the image of the actual transformation.
So we can also call this set right here the image of T. And now what is the image of T? Well, we know that we can write any-- and this is literally any-- so T is going from Rn to Rm. We can write T of x-- we can write any linear transformation like this-- as being equal to some matrix, some m by n matrix times a vector. And these vectors obviously are going to be members of Rn-- times sum Rn.
And what is this? So what is the image -- let me write it in a bunch of different ways -- what is the image of Rn under T? So we could write that as T -- let me write it this way. We could write that as T of Rn, which is the same thing as the image of T. Notice we're not saying under anything else, because now were saying the image of the actual transformation.
Which we could also write as the image of T. Well what are these equal to? This is equal to the set of all the transformations of x. Well all the transformations of x are going to be Ax where x is a member of Rn. So x is going to be an n-tuple, where each element has to be a real number.
So what is this? So if we write A-- let me write my matrix A. It's just a bunch of column vectors, a1, a2. It's going to have n of these, right? Because it has n columns. And so a times any x is going to be-- so if I multiply that times any x that's a member of Rn.
Tianlalu 4, 2 2 gold badges 18 18 silver badges 43 43 bronze badges. Add a comment. Active Oldest Votes. Tanner 1.
AxiomaticApproach AxiomaticApproach 1, 9 9 silver badges 19 19 bronze badges. Sign up or log in Sign up using Google.
Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. It seems that this problem is asking for another proof using the linearity conditions, but I don't see why such a proof is necessary.
So, not only is the proof I offer much simpler than writing the details your book provides, but it is also more general. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password. Post as a guest Name. Email Required, but never shown. Upcoming Events. Featured on Meta. Now live: A fully responsive profile.
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