Linear Algebra

Topic: Geometry of Linear Maps 271( xy)↦→( x2x + y↦−→)For contrast the next picture shows the effect of the map represented by C 2,1 (1).In this case, vectors are affected according to their second component. Thevector ( xy)is slid horozontally by twice y.( xy)↦→( x + 2yy↦−→)Because of this action, this kind of map is called a skew.With that, we have covered the geometric effect of the four types of componentsin the expansion H = T n T n−1 · · · T j BT j−1 · · · T 1 , the partial-identityprojection B and the elementary T i ’s. Since we understand its components,we in some sense understand the action of any H. As an illustration of thisassertion, recall that under a linear map, the image of a subspace is a subspaceand thus the linear transformation h represented by H maps lines through theorigin to lines through the origin. (The dimension of the image space cannotbe greater than the dimension of the domain space, so a line can’t map onto,say, a plane.) We will extend that to show that any line, not just those throughthe origin, is mapped by h to a line. The proof is simply that the partialidentityprojection B and the elementary T i ’s each turn a line input into a lineoutput (verifying the four cases is Exercise 6), and therefore their compositionalso preserves lines. Thus, by understanding its components we can understandarbitrary square matrices H, in the sense that we can prove things about them.An understanding of the geometric effect of linear transformations on R n isvery important in mathematics. Here is a familiar application from calculus.On the left is a picture of the action of the nonlinear function y(x) = x 2 + x. Asat that start of this Topic, overall the geometric effect of this map is irregularin that at different domain points it has different effects (e.g., as the domainpoint x goes from 2 to −2, the associated range point f(x) at first decreases,then pauses instantaneously, and then increases).5500