j 1 p the product C A ú B c ij i 1 m j 1 p is the matrix of size m p with

# J 1 p the product c a ú b c ij i 1 m j 1 p is the

• Notes
• 13

This preview shows page 4 - 7 out of 13 pages.

j =1: p , the product C = A ú B = [ c ij ] i =1: m , j =1: p is the matrix of size [ m , p ] with components c ik = n ÿ j =1 a ij b jk , for all i , k We typically omit the ú , and write C = AB for the matrix product. 3. Given A = [ a ij ] i =1: m , j =1: n and a complex number , the scalar multiple C = A is the matrix of the same size with components c ij = a ij , for all i , j Definition 1. 1. We have zero matrices O m , n of all sizes. 2. The identity matrix I n of size [ n , n ] (a “square” matrix) has components 1 on the diagonal, and 0 o ff the diagonal. 3. Some, but not all, square matrices A have an inverse A 1 such that A 1 A = AA 1 = I n : when the inverse exists, A is called an invertible matrix . Subscribe to view the full document.

1.2. VECTOR SPACES 7 4. The transpose of A = [ a ij ] i =1: m , j =1: n is the matrix A T = [ a T ij ] i =1: n , j =1: m with a T ij = ( a ji ) for all i , j . 5. The complex conjugate of A = [ a ij ] i =1: m , j =1: n is the matrix A ú = [ a ú ij ] i =1: m , j =1: n with a ú ij = ( a ij ) ú for all i , j . 6. The Hermitian conjugate of A is the matrix A = ( A T ) ú = ( A ú ) T . Another very important property of matrix algebra, often overlooked in elementary treatments, is that it respects block decompositions . For example, one can think of a [4, 6] matrix as a [2, 3] matrix whose components are [2, 2] matrices. Moreover, matrix multiplication can be performed in any way that respects such a block decomposition. We will see examples of this in the following discussion. 1.2 Vector Spaces We zoom straight to complex vector spaces. Definition 2. A set of objects ˛ a , ˛ b , ˛ c , . . . called vectors is called a complex vector space V if for all vectors ˛ a , ˛ b , ˛ c , . . . and scalars , , , . . . : 1. the set is closed under a vector addition with associativity ˛ a + ( ˛ b + ˛ c ) = ( ˛ a + ˛ b ) + ˛ c and commutativity ˛ a + ˛ b = ˛ b + ˛ a . 2. the set is closed under multiplication by scalars , , · · · œ C , which is associative ( —˛ a ) = ( –— ) ˛ a and distributive ( ˛ a + ˛ b ) = –˛ a + ˛ b , ( + ) ˛ a = –˛ a + —˛ a . 3. there is a zero vector ˛ 0 such that ˛ 0 + ˛ a = ˛ a . 4. the scalar 1 is such that 1 ˛ a = ˛ a . 5. all vectors ˛ a have a negative ˛ a such that ˛ a + ( ˛ a ) = ˛ 0 . Here are some important examples of complex vector spaces, and one example of a real vector space: Examples 1. 1. “Standard” C n , which we identify as the set of complex [ n , 1] matrices X (i.e. column n -vectors). 2. For each pair m , n of positive integers, the set of complex [ m , n ] matrices. 3. C [0, 1] , the set of continuous functions from the interval [0, 1] to C . 4. For each N , including N = Œ , W N , the set of complex trigonometric polynomials of the form f ( x ) = q N n = N a n e inx where a n œ C for each n . 5. “Standard” R n , the set of real [ n , 1] matrices X (i.e. column n -vectors), is likewise the first example of a real vector space , i.e a vector space taken with real scalars. 8 CHAPTER 1. LINEAR ALGEBRA (APPROX. 9 LECTURES) The central theme here is that many important examples exist which are not naturally a C n or R n . Subscribe to view the full document. • Winter '10
• kovarik

### What students are saying

• As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

Kiran Temple University Fox School of Business ‘17, Course Hero Intern

• I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

Dana University of Pennsylvania ‘17, Course Hero Intern

• The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

Jill Tulane University ‘16, Course Hero Intern