1. Lecture 1 Linear algebra Vectors, matrices

2. Linear algebra Encyclopedia Britannica:“a branch of mathematics that is concerned with mathematical structures closed under the operations of addition and scalar multiplication and that includes the theory of systems of linear equations, matrices, determinants, vector spaces, and linear transformations“ Deals with: – Vector – vector space – linear transformation of vectors – Matrix – system of linear equations – solution (Lecture 2)

3. Vectors in Two Dimensions Two-dimensional vectors can be defined as ordered pairs of real numers (a,b). a,b…components Vector is graphically represented by a directed line from the origin of a coordinate system to the point (a,b) whose length represents the magnitude and whose orientation in space represents the direction.

4. Basic operations

5. Illustration of addition and subtraction u v u+vu-v Addition: Place the two vectors tail-to-head and draw the resultant vector. The addition of vectors is commutative (parallelogram law). Rovnoběžník The parallelopiped is the analogue in space of the parallelogram in the plane.

6. Pythagorean theorem expressed in terms of vectors b a a b c = a+b c 2 = a 2 + b 2

7. Length, dot product Vectors v and w perpendicular, if their scalar product is equal to that are perpendicular to each other are orthogonal.The zero vector is perpendicular to all vectors. Example pp.194.

8. Multiple dimensional vectors A k-dimensional vector y is an ordered collection of k real numbers y 1, y 2,..., yk, and is written as y = (y 1, y 2,..., y k ). The numbers y j ( j = 1, 2,..., k) are called the components of the vector y. Example: u=(1, −3, 0, 5) is a four-dimensional vector. Its first component is 1, its second component is −3, and its third and fourth components are 0 and 5.

9. Transposition Row vector v = (v 1, v 2, …, v n ) Column vector

10. Scalar multiplication, addition Scalar multiplication: 4(3, 0,−1, 8) = (12, 0,−4, 32), vector addition: (3, 4, 1,−3) + (1, 3,−2, 5) = (4, 7,−1, 2). Using both operations, we can make the following type of calculation: (1, 0)x1 + (0, 1)x2 + (−3,−8)x3 = (x1, 0) + (0, x2) + (−3x3,−8x3) = (x1 −3x3, x2 −8x3). The dimension of vectors must be the same (2D, 3D, …).

11. Linear combination of vectors v 1, v 2, …, v n vectors a 1, a 2, …, a n scalars Linear combination of vectors v = a 1.v 1 + a 2.v 2 + … + a n.v n Result of this operation is a vector Sum of vectors multiplied by scalars

12. Linearly independent vectors Vectors v 1, v 2, …, v n are linearly dependent if there exist a 1, a 2, …, a n not all equal to zero, such that 0 = a 1.v 1 + a 2.v 2 + … + a n.v n Otherwise, these vectors are linearly independent

13. Matrices An m x n matrix is a rectangular array of numbers in m rows and n columns whose dimension is m by n. A is called square if m = n. The numbers a ij are referred to as the elements of A.

14. Row and column vectrors A k-by-1 matrix is called a column vector and a 1-by- k matrix is called a row vector. The coefficients in row i of the matrix A determine a row vector A i = (a i1, a i2, …, a in ), and the coefficients of column j of A determine a column vector A j = (a 1 j, a 2 j,..., a m j ). Column vectors are frequently written horizontally in angular brackets.

15. Matrices Main diagonal of matrix – Entries a 11, a 22,... a rr, for r = min(m,n) Diagonal matrix – Square matrix with non zero elements on the main diagonal Triangular matrix – Lower (left) and upper (right) triangular matrix Identity or unit matrix The identity matrix of order m, written I m (or simply I ) is a square m-by-m matrix with ones along the diagonal and zeros elsewhere.

16. Matrix operations Transpose Addition Scalar multiplication Matrix multiplication (product) Inversion Elementary row operations

17. Transpose The transpose of a matrix A, denoted A T, is formed by interchanging the rows and columns of A; that is, a T i j = a ji.

18. Addition Addition of two matrices A and B, both with dimension m by n, is defined as a new matrix C, written C = A + B, whose elements c ij are given by c ij = a ij + b ij. The usual matrix addition is defined for two matrices of same dimensions.

19. Scalar multiplication Scalar multiplication of a matrix A and a real number a is defined to be a new matrix B=a.A, whose elements b i j are given by: b i j = a.a i j

20. Matrix multiplication (product) The product of an m-by-p matrix A and a p-by-n matrix B is defined to be a new m-by-n matrix C, written C = A.B, whose elements c ij are given by: If the number of columns of A does not equal the number of rows of B, then AB is undefined.

21. Matrix multiplication Each element of the matrix product is a scalar product of vectors: row vector of A and column vector of B

22. Algebraic rules for matrices matrix multiplication is not commutative: AB ≠ B A A + B = B + A (Commutative law) A + (B + C) = (A + B) + C (Associative law) A(BC) = (AB)C (Associative law) A(B + C) = AB + AC (Distributive law)

23. Inversion of regular matrix Given a square m-by-m matrix B, if there is an m-by-m matrix D such that D.B = B.D = I D is called the inverse of B and is denoted B −1. properties of inverses: The inverse of a matrix B is unique if it exists. I −1 = I since I.I = I

24. Gauss–Jordan elimination System of linear equations: Bx = I y = y If B has an inverse, then multiplying on the left by B −1 yields: I x = B −1 y Goal of the elimination: one variable is isolated in each row.

25. Elementary row (column) operation Add one row to another row Multiplying row by a scalar Exchange the rows Releasing of the row The same operations can be provided on the columns

26. Example