Class Notes (836,067)
Mathematics (1,919)
MATH 136 (168)
Lecture 12

# Lecture 12.pdf

7 Pages
103 Views

School
Department
Mathematics
Course
MATH 136
Professor
Robert Sproule
Semester
Winter

Description
Friday, January 31 − Lecture 12 : Matrices and matrix algebra Concepts: 1. matrix and the operation of addition, multiplication, scalar multiplication and transposition on matrices. 2. rules of matrix algebra. 3. zero matrix 4. matrix-vector multiplication. 5. system of linear equations expressed as a matrix equation. 12.1 Definition − Matrices. If m and n are positive integers, a matrix of size m × n (or of dimension m × n), is a rectangular array of real numbers, arranged inm rows and n column. It is represented as A = [a ]ij m × n The a ’ijare called the entries orcomponents of the matrix. The symbol i is called the row index or the entrée and j is called its column index. Example : This is a 3 by 3 matrix A : 12.1.1 Two matrices are said to be equalmatrices if and only if their corresponding entries are equal. Because of this matrices of different dimensions cannot be equal. 12.1.2 Remark − A vector, such as u = (2, 1, 7) for example, can be also be expressed in matrix form, u = [2, 1, 7] 1 × 3, or The form it should take is usually determined by the context. 12.2 Definition − If A = [a ] ij m×n then the vectors r = ia , ai1 .i2, a ) iinℝ are called the n row-vectors of A (1 ≤ i ≤ m) . The vectors c j= (a 1ja 2j..., anj in ℝ are called the column-vectors of A (1 ≤ j ≤ n). So, when convenient, we can think of A as a pile of row- vectors, r , r , ..., r all stacked one on top the other, or as a juxtaposition of column 1 2 m vectors c ,1c 2 ..., cn: 12.3 Definitions − A matrix that contains only zeroes as entries is called the zero matrix. 12.4 Operations with matrices − Let M m×n denote the set of all m× n matrices with real numbers as entries. Then will define addition, scalar multiplicationand transposition on matrices in M m×n as follows: Operation What we start with : What we get : Addition Scalar multiplication Transposition Suppose then from the definition we see that the transpose of A exchange rows with columns: 12.4.1 Example – Let Aand B be the given matrices: T a) Compute 3A – 2B Answer: b) Find the (2, 3) entry of C = A + 3B. T Answer: Let C = [c ], ij= [b ], ij= A = [d ]. ij c23 d + 23 = a 233b =327 + 3(23 = −7 th T c) Find the (1, 3) entry of H = (2A + B) Answer: 4 th T d) Find the (2, 2) entry of E = 2A + A 12.4.2 Theorem – Let m and n be fixed positive integers. Then the set M m × nof all m × n matrices with real number entries equipped with + and scalar multiplication described above satisfiesthe following basic properties: 1) M is closed under addition and scalar multiplication. m × n 2) Addition in M is associative and commutative. m × n 3) Matrices in M distribute over finite sums of scalars. m × n 4) Scalars distribute over finite sums of matrices. 5) Transpose properties: i) (A ) = A, ii) (A + B) = A + B , T iii) (cA)T = cA T Proof : The proof of these follow immediately from the definition of + and scalar multiplication and transposition and so is omitted. 12.5 Some special matricesand matrix related definitions T  The square matrix A is said to be symmetric if A = A . That is, if A = [a ] ij n×nhen a ija .ji  A column matrix is a matrix which has only one column. While a row matrix is a matrix with only one row. Both a column matrix and row matrix can be referred to as vectors when useful and appropriate.  Given a square matrix A , tn×ntrace of the matrix
More Less

Related notes for MATH 136
Me

OR

Join OneClass

Access over 10 million pages of study
documents for 1.3 million courses.

Join to view

OR

By registering, I agree to the Terms and Privacy Policies
Just a few more details

So we can recommend you notes for your school.