Let's look at two more classic examples to make sure that we can generalize what we've learned. This is basically what eigenvectors and their corresponding eigenvalues are. These are our eigenvectors, and the value they are scaled by is know as an eigenvalue.įrom a conceptual perspective, that's about it for 2D eigen-problems, we simply take a transformation and we look for the vectors who are still laying on the same span as before, and then we measure how much their length has changed. So in some sense, the horizontal and vertical vectors are special, they are characteristic of this particular transformation. Lastly, the diagonal orange vector used to be exactly 45 degrees to the axis, but it's angle has now increased as has its length.īesides the horizontal and vertical vectors, any other vectors' direction would have been changed by this vertical scaling. The vertical pink vector is also still pointing in the same direction as before but its length is doubled. Now, consider our vertical scaling again, and think about what will happen to these three vectors.Īs you can see, the horizontal green vector is unchanged, i.e., it is pointing in the same direction and having the same length. To highlight this, lets draw three specific vectors onto our initial square. Notice that, after the transformation is applied, some vectors end up lying on the same line that they started on whereas, others do not. Whereas, if we applied a horizontal shear to this space, it would become a trapezoid: For example, if we apply a scaling of 2 in the vertical direction, the square would become a rectangle. This is most easily visualized by drawing a square centered at the origin, and then observing how the square is distorted when you apply the transformation. However, it can also be useful to think about what it might look like when they are applied to every vector in this space. Often, when applying these transformations, we are thinking about what they might do to a specific vector. These operations can include scalings, rotations, and shears. But once you know how to sketch these problems, the rest is just algebra.Īs you've seen from previous weeks, it's possible to express the concept of linear transformations using matrices. This topic is often considered by students to be quite tricky.
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