[...]... Vector with the elements filled up with real numbers Œ2:89 21:87 100 Column Vector with the elements filled up with Complex numbers 2 3 1Cj 6 j 7 6 7 49 C 7j 5 0 Matrix of size 2 3 with the elements filled up with real numbers Matrix of size 3 2 3 # 6 4 " 1 2 2 with the elements filled up with complex numbers 2 j 6 6 2j 4 0 1Cj 3 7 5j 7 5 j E.S Gopi, Mathematical Summary for Digital Signal Processing Applications. .. represented with respect to the standard basis Also note that the transformation matrix is with respect to the standard basis R2 Vector points plotted in the 2D plane before and after transformation are given below (Fig 1.2) vector [x y] 2D Map before Transformation vector [−x −y] 2D Map after Transformation Fig 1.2 Illustration of the linear transformation of the vector 1.12 Transformation Matrix with Different... Compute the Transformation Matrix Ä Ä 1 1 Identify and note down the transformed vectors for the standard basis and 0 0 Ä Ä 1 0 For the example mentioned above and are the transformed vectors 0 1 Ä Ä 1 1 corresponding to the standard basis and respectively The transformation 0 0 matrix corresponding to the above transformation is obtained by representing the transformed vectors column wise For the above... T ŒA  ŒCT  ŒBT  ŒDT  (h) Square matrix The matrix with number of Rows is equal to the number of Columns (i) Identity matrix The square matrix with all the elements is filled up with zeros except the diagonal elements which are filled up with all ones 6 1 Matrices (j) Lower triangular matrix The square matrix with all the elements is filled up with zeros except the elements in the diagonal and below... column wise For the above mentioned example, the transformation matrix is given as Ä 1 0 0 1 This is same as the one given in the Example 1.14 1.12 Transformation Matrix with Different Basis Consider the transformation matrix with respect to the standard basis as described below Ä 1 0 0 1 Ä Ä Ä 5 2 32 3 1 1 5 2 32 3 Consider the vector D represented with re6 4154 1 5 1 1 6 415;405 1 1 0 1 Ä Ä Ä Ä 1 1... matrix is the matrix with zeros in the upper triangular portion of the matrix with at least one non-zero element in the remaining portion (k) Upper triangular matrix The square matrix with all the elements is filled up with zeros except the elements in the diagonal and above the diagonal which are filled up at least one non-zero elements In other words, Upper triangular matrix is the matrix with zeros in the... zeros in the Lower triangular portion of the matrix with at least one non-zero element in the remaining portion (l) Diagonal matrix The square matrix with all the elements is filled up with zeros except the diagonal elements which are filled up with at least one non-zero element in the diagonal (m) Permutation matrix Permutation matrix is one when multiplied with the matrix interchanges the elements of the... element z 2 V such that z C x D x for all x 2 V z is called zero vector (d) Additive inverse For each x 2 V , there exists y 2 V such that x C y D z (e) There exists 1 2 F , such that 1:x D x for all x 2 V (f) For all a; b 2 F and x 2 V a:.b:x/ D ab/:x (g) For all a 2 F and x; y 2 V a:.x C y/ D a:x C a:y (h) For all a; b 2 F and x 2 V a C b/:x D a:x C b:y 3 Subspace S of the vector space V is a subset... (Ä Ä 5 2 32 3 1 1 Similarly D 6 415405 1 1 0 1 1 Ä 5 6 ) 2 32 1 1 4 5;4 3 15 1 1.11 Linear Transformation of the Vector T: V->U is the Linear transformation such that any vector in the vector space ‘V’ is mapped to another vector that lies in the vector space ‘U’ There exists the transformation matrix to perform this operation The vector space V can also be equal to the vector space U Example 1.14 T:... 1.10 Vector Representation with Different Basis Example 1.13 Ä 3 4 2 32 3 1 0 0 1 4 54 5 2 R2 is the vector represented with respect to the basis Ä Ä Ä 1 0 3 (i.e.) The vector 0 1 4 Ä 2 32 3 1 0 0 1 4 54 5 D3 1 0 Ä 0 C4 15405 1 4 0 1 2 32 3 2 32 3 1 0 0 1 4 54 5 1.10 Vector Representation with Different Basis Ä 5 Similarly 6 2 32 1 1 4 54 3 2 R2 is the vector represented with respect to the basis 15 . Mathematical Summary for Digital Signal Processing Applications with Matlab E.S. Gopi Mathematical Summary for Digital Signal Processing Applications with. my wife G. Viji Preface The book titled Mathematical summary for Digital Signal Processing Applications with Matlab consists of Mathematics which is