2/19/2002, LINEAR TRANSFORMATIONS (III) Math 21b, O. KnillHOMEWORK: 2.4: 4,14,26*,28,40,48MATRIX PRODUCT. If B is a m ×n matrix and A is a n ×p matrix, then BA is a m ×p matrix with entries(BA)ij=Pnk=1BikAkj.EXAMPLE. If B is a 3 × 4 matrix, and A is a 4 × 2 matrix then BA is a 3 × 2 matrix.B =1 3 5 73 1 8 11 0 9 2, A =1 33 11 00 1, BA =1 3 5 73 1 8 11 0 9 21 33 11 00 1=15 1314 1110 5.COMPOSING LINEAR TRANSFORMATIONS. If S : Rm→ Rn, x 7→ Ax and T : Rn→ Rp, x 7→ Bxare linear transformations, then their composition T ◦ S is a linear transformation from Rmto Rp. Thecorresponding matrix is the matrix product BA.EXAMPLE. Find the matrix which is a composition of a rotation around the x-axes by π/2 followed by arotation around the y-axes by −π/2.SOLUTION. The first transformation has the property that e1→ e1, e2→ −e3, e3→ e2, the second e1→−e2, e2→ e1, e3→ e3. If A is the matrix belonging to the first transformation and B the second, then BA isthe matrix to the composition.B =0 1 0−1 0 00 0 1A =1 0 00 0 10 −1 0, BA =0 0 1−1 0 00 −1 0. The composition maps e1→ −e2→ e3→ e1is a rotation around a long diagonal.EXAMPLE. A rotation dilation is the composition of a rotation by α = arctan(b/a) and a scale by r =√a2+ b2.WHY? Matrix multiplication is a generalisation of usual multiplication of numbers or the dot product.PROPERTIES. Note that AB 6= BA in general! Otherwise, the same rules apply as for numbers: A(BC) =(AB)C, AA−1= A−1A = 1n, (AB)−1= B−1A−1, A(B + C) = AB + AC, (B + C)A = BA + CA etc.PARTITIONED MATRICES. The entries of matrices can themselves be matrices. If B is a m × n matrix andA is a n × p matrix, and assume the entries are k × k matrices, then BA is a m × p matrix where each entry(BA)ij=Pnk=1BikAkjis a k ×k matrix. Partitioning matrices can improve matrix multiplication (i.e. Strassenalgorithm).EXAMPLE. If A =A11A12A00A22, where Aijare k × k matrices with the property that A11and A22areinvertible, then B =A−111−A−111A12A−1220 A−122is the inverse of A.1423NETWORKS. Let us associate to the computer network (shown at the left) amatrix0 1 1 11 0 1 01 1 0 11 0 1 0To a worm in the first computer we associate a vector1000. The vector Ax has a 1 at the places, where the worm could be in the nextstep. The vector (AA)(x) tells, in how many ways the worm can go from the firstcomputer to other hosts in 2 steps. In our case, it can go in three different waysback to the computer itself.Matrices help to solve combinatorial problems (see movie ”Good will hunting”).For example, what does [A1000]22tell about the worm infection of the network?What does it mean if A100has no zero entries?FRACTALS. Closely related to linear maps are affine maps x 7→ Ax + b. They are compositions of alinear map with a translation. It is not a linear map if B(0) 6= 0. Affine maps can be disguised as lin-ear maps in the following way: let y =x1and the (n+1)∗(n+1) and B =A b0 1, then By =Ax + b1.Fractals can be constructed by taking for example 3 affine maps f, g, h which contract area. For a given objectY0define Y1= f(Y0) ∪g(Y0) ∪h(Y0) and recursively Yk= f(Yk−1) ∪g(Yk−1) ∪h(Yk−1). Above you see Ykaftersome iterations. In the limit, Ykbecomes a fractal, an object with noninteger dimension.CHAOS. Consider a map in the plane like T :xy7→2x + 2 sin(x) − yxWe apply this map again andagain, we look at points (x1, y1) = T (x, y), (x2, y2) = T (T (x, y)), etc. One writes Tnfor the n-th iteration ofthe map and (xn, yn) for the image of (x, y) under the map Tn. The linear approximation of the map at apoint (x, y) is the matrix DT (x, y) =2 + 2 cos(x) − 11. (If Txy=f(x, y)g(x, y), then the row vectors ofDT (x, y) are just the gradients of f and g). T is called chaotic at (x, y), if the entries of D(Tn)(x, y) growexponentially fast with n. By the chain rule, D(Tn) is the product of matrices DT (xi, yi). For example, T ischaotic at (0, 0). If there is a positive probability to hit a chaotic point, then T is called chaotic.FALSE COLORS. Any color can be represented as a vector (r, g, b), where r ∈ [0, 1] is the red g ∈ [0, 1] is thegreen and b ∈ [0, 1] is the blue component. Changing colors in a picture means applying a transformation on thecube. Let T : (r, g, b) 7→ (g, b, r) and S : (r, g, b) 7→ (r, g, 0). What is the composition of these two linear maps?OPTICS. Matrices help to calculate the motion of light rays through lenses. Alight ray y(s) = x + ms in the plane is described by a vector (x, m). Followingthe light ray over a distance of lenght L corresponds to the map (x, m) 7→(x + mL, m). In the lense, the ray is bent depending on the height x. Thetransformation in the lense is (x, m) 7→ (x, m − kx).xm7→ ALxm=1 L0 1xm,xm7→ Bkxm=1 0−k 1xm.Examples:1) Eye of length R looking far: ARBk. 2) Eye of length R looking at distance L: ARBkAL. 3)
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