Linear Algebra

Topic: Geometry of Linear Maps 273If 0 k < 1 or if k < 0 then the i-th component goes the other way, here to theleft.( ( )x −2x↦→y)y−−−−−−−−→Either of these is a dilation.A transformation represented by a P i,j matrix interchanges the i-th and j-thaxes. This is reflection about the line x i = x j .( (x y↦→y)x)−−−−−−−→Permutations involving more than two axes decompose into a combination ofswaps of pairs of axes; see Exercise 5.The remaining matrices have the form C i,j (k). For instance C 1,2 (2) performs2ρ 1 + ρ 2 .(xy) ( )1 02 1E 2 ,E 2−−−−−−−−→x2x + y( )In the picture below, the vector ⃗u with the first component of 1 is affected lessthan the vector ⃗v with the first component of 2 — h(⃗u) is only 2 higher than ⃗uwhile h(⃗v) is 4 higher than ⃗v.h(⃗u)h(⃗v)⃗u⃗v( ( )x x↦→y)2x + y−−−−−−−−−−→Any vector with a first component of 1 would be affected in the same way as⃗u; it would slide up by 2. And any vector with a first component of 2 wouldslide up 4, as was ⃗v. That is, the transformation represented by C i,j (k) affectsvectors depending on their i-th component.Another way to see this point is to consider the action of this map on the unitsquare. In the next picture, vectors with a first component of 0, like the origin,are not pushed vertically at all but vectors with a positive first component slideup. Here, all vectors with a first component of 1, the entire right side of thesquare, slide to the same extent. In general, vectors on the same vertical lineslide by the same amount, by twice their first component. The shape of theresult, a rhombus, has the same base and height as the square (and thus thesame area) but the right angle corners are gone.( ( )x x↦→y)2x + y−−−−−−−−−−→