Linear maps on Rn Linear structure of Rn n I The vector space R has a linear structure with two features: n n I Vector addition: For all u; v 2 R we have u + v 2 R with the components (u + v)i = ui + vi for 1 ≤ i ≤ n with respect, say, to the standard basis vectors of Rn. n I Scalar multiplication: For all u 2 R and all c 2 R, we n have cu 2 R with the components (cu)i = cui for 1 ≤ i ≤ n. I Note that these two structures exist independent of any particular basis we choose to represent the coordinates of vectors. cu u+v v u u Convex combination of vectors 2 I The convex combination of any two vectors u; v 2 R is the line segment au + bv joining u and v, where 0 ≤ a ≤ 1, 0 ≤ b ≤ 1 and a + b = 1. v u I Similarly, the convex combination of three vectors u; v; w 2 R3 is the triangle au + bv + cw with vertices u, v and w, where 0 ≤ a; b; c ≤ 1 and a + b + c = 1. w v u Scalar product of vectors 3 I Given two vectors u; v 2 R , with coordinates 2 3 2 3 u1 v1 u = 4u25 v = 4v25 u3 v3 their scalar product is given by 3 X u · v = ui vi = kukkvk cos θ; i=1 q 2 2 2 where kuk = u1 + u2 + u3 and θ is the angle between u and v. Check this formula in R2! v u θ Scalar product and coordinates I Note that the lengths kuk and kvk as well as the angle θ are independent of the three mutually perpendicular axes used to determine the coordinates of u and v. I It follows that if we move to another set of mutually perpendicular axes with respect to which u and v have coordinates 2 0 3 2 0 3 u1 v1 0 0 u = 4u25 v = 4v25 ; 0 0 u3 v3 then, we have 3 3 X X 0 0 u · v = ui vi = kukkvk cos θ = ui vi : i=1 i=1 Linear maps n m I A map f : R ! R is called linear if it preserves the linear structure, i.e., if it satisfies: n I 8u; v 2 R : f (u + v) = f (u) + f (v). n I 8u 2 R 8c 2 R: f (cu) = cf (u). I We can write the two conditions as a single condition: n I 8u; v 2 R 8a; b 2 R: f (au + bv) = af (u) + bf (v). n n I Example. For a fixed u 2 R , let fu : R ! R be defined by fu(v) = u · v I It is easy to check that fu is a linear map. I If kuk = 1, then fu(v) is the projection of v onto the direction of u. v u ||u||=1 u.v Linear maps and basis n m I Let f : R ! R be a linear map. n I Assume e1; e2;:::; en form a basis of R and d1; d2;:::; dm form a basis of Rm. I By linearity, the action of f is completely determined by m knowing the vectors f (ej ) 2 R (j = 1; ··· n). m I Since d1;:::; dm form a basis of R , each vector m f (ej ) 2 R can be expressed as m X f (ej ) = Aij di ; i=1 for some coefficients Aij 2 R with i = 1; ··· ; m. Pn I Given any vector v = j=1 vj ej , we have by linearity: n n n m X X X X f (v) = f ( vj ej ) = vj f (ej ) = vj ( Aij di ) j=1 j=1 j=1 i=1 Matrix representation of linear maps I The m × n matrix matrix A = (Aij ) with 1 ≤ i ≤ m; 1 ≤ j ≤ n is called the matrix representation of f with respect to n m the basis e1;::: en of R and the basis d1;:::; dm of R . Pn I Given any vector v = j=1 vj ej , we can compute the vector representing f (v) 2 Rm with respect to the basis d1;:::; dm: I n n n m X X X X f (v) = f ( vj ej ) = vj f (ej ) = vj ( Aij di ) j=1 j=1 j=1 i=1 n m m n m X X X X X = Aij vj di = Aij vj di = (Av)i di i=1 j=1 i=1 j=1 i=1 I Thus, (Av)i is the coefficient of di in the expansion of f (v) with respect to the basis d1;:::; dm, i.e, I Av represents f (v) with respect to the basis d1;:::; dm. Extension to Cn n I All the definitions and results for R we have seen here can be extended to the vector space Cn, the complex vector space of dimension n. 2 T I For example, C has vectors of the form v = (v1; v2) with v1; v2 2 C. n n I The linear structure of C is like R , except that we can now use scalar multiplication with complex numbers, i.e., T for the vector v above: cv = (cv1; cv2) for c 2 C. n I Interestingly, the standard basis for R can be interpreted as a standard basis for Cn. I Note that even when we deal with real matrices, we cannot avoid using complex vectors. I In fact, the eigenvalues of a real matrix can be complex numbers, which means that the corresponding eigenvectors would be complex vectors. The inner product in Cn n I How about the dot product and the norm of a vector in C ? I If we simply copy the definition of the dot product from the real case for complex vectors as well, then the dot product of a complex vector with itself will not necessarily be a non-negative number and thus cannot be used to define the norm of the vector. n I For complex vectors v; w 2 C , we define the inner Pn ∗ product hv; wi := i=1 vi wi . I Recall that for c = c1 + ic2 2 C the complex conjugate of ∗ c is given by c = c1 − ic2. ∗ 2 2 I Note that c c = c1 + c2 ≥ 0 for any c = c1 + ic2 2 C. I Observe that the inner product is not commutative: hv; wi = hw; vi∗. n p I The norm of v 2 C is defined as kvk := hv; vi..