i.e. the i-th column of A is the value of T at the i-th standard basis vector e i R n.
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1 Aim lecture In first year you learnt that you can mutliply not only a (real matrix with a (real vector, but more generally, matrices together (of compatible sizes. Furthermore, you learnt the following matrix representation theorem which essentially states that linear maps from R n R m are no more complicated than matrices. First Year Matrix Representation Theorem Let T : R n R m be a linear map. Then there is a unique matrix A M mn (R such that T is left multliplication by A, i.e. T v = Av for all v R n. In fact A = (T e 1 T e 2... T e n i.e. the i-th column of A is the value of T at the i-th standard basis vector e i R n. Aim lecture: Generalise matrix multiplication to matrices of linear maps and also the matrix representation thm above. Daniel Chan (UNSW Lecture 12: Matrix multiplication. Representation theorem. Semester / 9
2 Matrix multiplication Consider direct sums of F-spaces U = l k=1 U k, V = m j=1 V j, W = n i=1 W i. Prop-Defn Consider matrices T = (T ij ij (L(V j, W i ij, S = (S jk jk (L(U k, V j jk. We define the product matrix TS (L(U k, W i ik to be the one with (i, k-th entry (TS ik = m T ij S jk. The linear map assoc to TS is the composite map T S : U W. j=1 Proof. Consider (u k k U. Then we calculate (T S(u k k = T [S(u k k ] = T [( k = ( j T ij [ k S jk u k j ] S jk u k ] i = ( j T ij S jk u k i = (TS(u k k k Daniel Chan (UNSW Lecture 12: Matrix multiplication. Representation theorem. Semester / 9
3 Examples of matrix multiplication E.g Compute the product ( d ( dx 0 d 2 d dx 3 d dx id 0 dx 2 id d d dx dx + id = Fact Products of matrices of matrices are the same as the product of the big matrices of scalars. This follows since they both correspond to the same composite of linear maps. Daniel Chan (UNSW Lecture 12: Matrix multiplication. Representation theorem. Semester / 9
4 More examples E.g. Let I be the 2 2 identity matrix & A be the 4 4-matrix ( 0 I I 0. Find all 4 4-matrices X such that AX = XA. Daniel Chan (UNSW Lecture 12: Matrix multiplication. Representation theorem. Semester / 9
5 Matrix representation theorem Consider direct sums of F-spaces V = n j=1 V j, W = m i=1 W i. We have seen an example where we were given a linear map from V W and we wrote it as a matrix in (L(V j, W i ij. This is always possible by the following Theorem The map Φ : (L(V j, W i ij L(V, W which assigns to a matrix T = (T ij ij the associated linear map v T v is an isomorphism of vector spaces. In particular, every linear map from V W can be written uniquely in the form of a matrix in (L(V j, W i ij. Proof. Occupies next two slides & will only be sketched. Daniel Chan (UNSW Lecture 12: Matrix multiplication. Representation theorem. Semester / 9
6 Proof of bijectivity of Φ We first prove the map Φ is surjective, i.e. any linear map T : V W comes from a matrix. We will only look at the case m = 1, the general case (ex follows by noting that an m n-matrix is just a length n row matrix of length m column matrices. To keep notn simple, we will also assume n = 2 though the general case is just as easy & just requires lengthier notn. Hence T : V 1 V 2 W. Recall natural isomorphisms Ψ 1 : V 1 V (1 = V 1 0 : v 1 ( v 1 0, Ψ2 : V 2 V (2 = 0 V 2 : v 2 ( 0 v 2. Suffice prove T = Φ(T 1 T 2 where Hence Φ(T 1 T T 1 = T Ψ 1 : V 1 W, T 2 = T Ψ 2 : V 2 W ( v1 = T v 2 ( ( v1 0 + T 0 v2 = T 1 v 1 + T 2 v 2 = ( T 1 T 2 ( v 1 v 2 T 2 = T & surjectivity is proved. = T (Ψ 1 v 1 + T (Ψ 2 v 2. Daniel Chan (UNSW Lecture 12: Matrix multiplication. Representation theorem. Semester / 9
7 Proof injectivity of Φ We now prove that Φ is injective. Suppose then that Φ(T 1 T 2 = Φ(T 1 T 2 for some linear maps T 1, T 1 : V 1 W & T 2, T 2 : V 2 W. Suffice check T 1 = T 1, T 2 = T 2 by showing they have the same outputs for every possible input. Let v 1 V 1. Then ( ( v1 T 1 v 1 = (T 1 T 2 = (T 1 T v1 0 2 = T 0 1v 1. Hence T 1 v 1 = T 1 v 1 for all v 1 V 1 which shows T 1 = T 1. Sim (ex one sees T 2 = T 2. This completes the proof of injectivity & hence bijectivity. Daniel Chan (UNSW Lecture 12: Matrix multiplication. Representation theorem. Semester / 9
8 Proof linearity of Φ Recall first that the vector space structure of (L(V j, W i ij is the direct sum of the L(V j, W i. We need to check that for any (T ij, (T ij (L(V j, W i ij, β F and v V we have Φ((T ij + (T ijv = [Φ(T ij + Φ(T ij]v, Φ(β(T ij v = [βφ(t ij ]v. This follows from the following distributive & associative laws of the matrix-vector product Prop With the above notn 1 [(T ij + (T ij ]v = (T ijv + (T ij v 2 [β(t ij ]v = β[(t ij v] Proof. This is an easy check. We do 2 as an example. Write v = (v j j. Then [β(t ij ]v = (βt ij (v j j = ( j βt ij v j i = β( j T ij v j i = β[(t ij v]. Daniel Chan (UNSW Lecture 12: Matrix multiplication. Representation theorem. Semester / 9
9 Special case of row matrices Cor Let T : F n V be a linear map of F-spaces. Then T v = Av where A V n is the row matrix A = (T e 1... T e n i.e. the i-th column of A is T e i where e i is i-th standard basis vector. Proof. Thm = T is represented by left multn by some A V n and we deduce the i-th column of A must be Ae i = T e i E.g. Represent the linear map T : R 3 M 22 (R below by a row matrix (of matrices!. β 1 ( T β 2 β1 + β = 2 β 3. β 3 β 2 β 3 β 3 Daniel Chan (UNSW Lecture 12: Matrix multiplication. Representation theorem. Semester / 9
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