MATH136 Lecture Notes - Lecture 12: Row And Column Vectors, Scalar Multiplication, Linear Combination
14 May 2018
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Department
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Friday, May 26
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Lecture 12 : Matrices and matrix algebra (Refers to section 3.1)
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.
6. matrix times a column vector as a linear combination of column vectors
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 in m rows and n
column. It is represented as A = [aij]m × n
The aij’s are called the entries or components 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 equal matrices 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 = [aij]m×n, then the vectors ri = (ai1, ai2, ..., ain) in ℝn are called the
row-vectors of A (1 ≤ i ≤ m) . The vectors cj = (a1j, a2j, ..., amj) in ℝm are called the
column-vectors of A (1 ≤ j ≤ n). So, when convenient, we can think of A as a pile of row-
vectors, r1, r2, ..., rm all stacked one on top the other, or as a juxtaposition of column
vectors c1, c2, ..., cn :
12.3 Definitions − A matrix that contains only zeroes as entries is called the zero matrix.
12.4 Operations with matrices − Let Mm×n denote the set of all m × n matrices with real
numbers as entries. Then will define addition, scalar multiplication and transposition on
matrices in Mm×n as follows:
Suppose
Operation What we start with : What we get :
Addition
Scalar
multiplication
Transposition
AT is in Mn×m
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Document Summary
Friday, may 26 lecture 12 : matrices and matrix algebra (refers to section 3. 1) 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 in m rows and n column. The aij"s are called the entries or components of the matrix. The symbol i is called the row index or the entr e 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 equal matrices 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.
