Linear Algebra

Topic: Geometry of Linear Maps 273The first map f dilates the neighborhood of x by a factor ofdfdx (x)and the second map g dilates some more, this time dilating a neighborhood off(x) by a factor ofdgdx ( f(x) )and as a result, the composition dilates by the product of these two.In higher dimensions the map expressing how a function changes near apoint is a linear map, and is expressed as a matrix. (So we understand thebasic geometry of higher-dimensional derivatives; they are compositions of dilations,interchanges of axes, shears, and a projection). And, the Chain Rule justmultiplies the matrices.Thus, the geometry of linear maps h: R n → R m is appealing both for itssimplicity and for its usefulness.Exercises1 Let h: R 2 → R 2 be the transformation that rotates vectors clockwise by π/4 radians.(a) Find the matrix H representing h with respect to the standard bases. UseGauss’ method to reduce H to the identity.(b) Translate the row reduction to to a matrix equation T jT j−1 · · · T 1H = I (theprior item shows both that H is similar to I, and that no column operations areneeded to derive I from H).(c) Solve this matrix equation for H.(d) Sketch the geometric effect matrix, that is, sketch how H is expressed as acombination of dilations, flips, skews, and projections (the identity is a trivialprojection).2 What combination of dilations, flips, skews, and projections produces a rotationcounterclockwise by 2π/3 radians?3 What combination of dilations, flips, skews, and projections produces the maph: R 3 → R 3 represented with respect to the standard bases by this matrix?( ) 1 2 13 6 01 2 24 Show that any linear transformation of R 1 is the map that multiplies by a scalarx ↦→ kx.5 Show that for any permutation (that is, reordering) p of the numbers 1, . . . , n,the map⎛ ⎞ ⎛ ⎞x 1 x p(1)x 2x p(2)⎜⎝⎟⎠ ↦→ ⎜⎝⎟.⎠.x n x p(n)can be accomplished with a composition of maps, each of which only swaps a singlepair of coordinates. Hint: it can be done by induction on n. (Remark: in the fourth