Samenvatting Lineaire algebra 1
Lineaire Algebra (AESB1311)
Technische Universiteit Delft
Aanbevolen voor jou
Reacties
Preview tekst
Lineaire algebra 1 Chapter 1 Paragraph 1 System of lineair equations Linear equations: are in the same form Linear system: a collection of one or more linear equations involving the same variables. Solution set: set of all possible solutions Equivalent: when two linear systems have the same solution set. A system of linear equations is: Consistent: One solution Infinitely many solution Inconsistent: No solutions Coefficient matrix: coefficients of each variables of a linear system aligned in columns augmented matrix Size: tells how many rows and columns a matrix has: m x n m rows n columns Row equivalent: when there is a sequence of elementary row operations that transform one matrix into the other, if the augmented matrices are row equivalent they have the same solution set. Elementary row operations: Replacement Interchange Scaling Row operations are invertible Two fundamental questions: 1. Is the system consistent, that is, does at least one solution exist? 2. If a solution exists, is it the only that is, is the solution unique? Paragraph 2 Row reduction and echelon forms Leading entry: leftmost nonzero entry of a row. Echelon matrix: matrix which is in echelon form, matrix Matrix is in echelon form if it has the following properties: 1. All zero rows cover the lowest rows of the matrix. 2. Each leading entry is to the right of the leading entry of the row above it. 3. All entries in a column below a leading entry are zero. Matrix is in reduced echelon form when: 4. The leading entry in each nonzero row is 1. 5. Each leading 1 is the only nonzero entry in its column. A pivot position is the position of the leading 1 in a reduced echelon form. A pivot column is a column that contains a pivot position. Basic variables: defined (columns with pivot position). Free variables: undefined (no pivot position in column) any value for variable. Paramedic descriptions: solution of a system with free variables. Paragraph 3 Vector equations (Column) vectors: ordered lists of numbers, matrix with only one column. Rn: R stands fort he real numbers that appear as entries in a vector, n indicates that each vector contains n entries. Equal: two vectors in Rn are equal if only if their corresponding entries are equal. Scalar: the number c in cu, vector cu is contained multiplying each entry in u c. Geometric description: geometric point (a, b) can be identified as vector Parallelogram rule for addition: if u and v in R2 are represented as points in a plane, than is the fourth vertex of the parallelogram whose other vertices are u, v and 0. If v1, vp are in Rn, then the set of all linear combination of v1, vp is denoted and is called subjet of Rn spannend opspansel v1, vp. Geometric description of If u and v are nonzero vectors in R3, with v not a multiple of u, then the is the plane R 3 that contains u, v, 0. Paragraph 4 The matrix equation If A is an m x n matrix, with columns an and if x is in Rn, then the product of A and x is the linear combination of the columns of a using the corresponding entries in x as Ax a2 x1a1 x2a2 xnan Note that Ax is defined only if the number of columns of A number entries of x The equation of has a solution if b is a linear combination of the columns A. Let A be an m x n matrix. Then the following statements are logically equivalent: a. For each b in Rm, the equation has a solution. b. Each b in Rm is a linear combination of the columns of A. c. The columns of A span Rm. d. A has a pivot position in every row. Paragraph 5 Solution sets of linear systems Homogeneous: A linear system which can be written in the form always at least one solution, namely trivial solution. The homogeneous equation has a nontrivial solution if only if the equation has one free variable. Parametric Vector Equation: x su tv. Parametric Vector Form: explicit description of a plane with vectors. Suppose the equation is consistent for some given b, and let p be a solution. Then the solution set of is the set to all vectors of the form w p v, where v is any solution of the homogenous equation Paragraph 7 Linear independence Linear independent if only the trivial solution (no free variable) Linear dependent if nontrivial solutions (free variable) A set of two vectors is linear dependent if at least one of the vectors is a multiple of the other. A set of two vectors is linear independent if neither is a multiple of the other. Characterization of linearly dependent sets: An indexed set v1, of two or more vectors is linearly dependent i.o. at least one of the vectors in S is a linear combination of the other. A set is automatic linearly dependent when there are more vectors than entries in Rn with p n. If S contains the zero vector, then the set is linearly dependent. Chapter 2 Paragraph 2 Matrix operations Scalar entry in the ith row and jth column a ij: i row j column. Diagonal matrix is nxn whose nondiagonal entries are 0. Ex. nxn identity matrix In. A B is only defined if A and B are equal (same size). If A is an mxn matrix, and if B is an nxp matrix with columns b1, bp, then the product AB is the mxp matrix whose columns are Ab1, Abp AB b2 ... Ab2 Rowi(AB) rowi(A) B WARNINGS 1. In general, AB BA. 2. If AB AC, then it is not true in general that B C. 3. If AB 0, you cannot conclude or Transpose of mxn A nxm matrix A, denoted A T, whose columns are formed the former rows of A. a. (AT)T A b. AT BT c. (AB)T BTAT Paragraph 2 The inverse of a matrix 1 d a b If ad bc 0, then A is invertible and . If ad bc c d a 0, then A is not invertible. Determinant: ad bc, detA ad thus 2x2 matrix is invertible as long detA 0. If A is invertible: Ax b has the unique solution x A A Paragraph 2 Characterizations of invertible matrices Let A be a nxn matrix. Then the following statements are equivalent. a. A is an invertible matrix. b. A is row equivalent to nxn identity matrix. c. A has n pivot points. d. The equation has only the trivial solution. e. The columns of A form a linearly independent set. f. The linear transformation x Ax is g. The equation has at least one solution for each b in R n. h. The columns of A span Rn. i. The linear transformation x Ax maps Rn onto Rn. j. There is an nxn matrix C such that k. There is an nxn matrix D such that l. AT is an invertible matrix. If then A and B are both invertible with and Let T : Rn Rn be a linear transformation and let A bet he standard matrix for T. Then T is invertible i.o. A is invertible Paragraph 2 Expansion If A is mxn and B is nxp, then AB row 1(B) col1(A)row1(B) coln(A)rown(B). row 2(B) Paragraph 2 Matrix factorization Factorization of a matrix A: equation that expresses A as a product of two or more matrices. If A mxn and you can make an echelon form without row interchanges rijoptellingen naar that echelon form is U. When U is an echelon form of A, than L will exist as a mxn matrix z.d.: a. b. lower triagular matrix with 1s on the diagonal. How to do this: a. First operate matrix A in an echelon form U with only naar b. If u exists than operate Im to L doing the inverse steps done a in backwards order. Why are L and U useful? When A LU, the equation can be written as Writing y for Ux, we can find x solving the pair of equations. 1st. makes y, 2nd fill in y to find x: Paragraph 2 Subspaces of Rn Subspaces: important sets of vectors in R n. provide useful information about A subspace of Rn is any set of H in Rn that has three properties: a. The zero vector is in H. b. For each u and v in H, the sum is in H. c. For each u in H and scalar c, the vector cu is in H. The column space of a matrix A is the set ColA of all linear combination of the columns of A. The null space of a matrix A is the set of NulA of all solutions of the homogeneous equation A basis for a subspace H of Rn is a linearly independent set in H that spans H. The pivot columns of a matrix A form a basis for the column space of A. Paragraph 2 Dimension and Rank Suppose the set B is a basis for a subspace H. For each x in H, the coordinate of x relative to the basis B are weights c1, cp c1 such that cpbp and the vector in Rp is called the cp coordinate vector of x (relative to B) or the vector of x. Paragraph 3 rule, volume and linear transformation rule: Let A be an invertible nxn matrix. For any b in R n, the unique solution x of has entries given x det Ai (b) detA Inverse Formula: let a be an invertible nxn matrix. Then A A Adjugate of 1 adjA detA C 11 C 21 (numbers for columns and rows have switched). C 12 C 22 i column, j row The area of an object is detA. If S is a parallelogram in 2x2 matrix A in R 2, then: of of If S is a parallelepiped in R3 with matrix A 3x3 then, of of R2 area
Samenvatting Lineaire algebra 1
Vak: Lineaire Algebra (AESB1311)
Universiteit: Technische Universiteit Delft
- Ontdek meer van:
