Linear Algebra Theorems and Definitions, Exams of Nursing

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Linear Algebra and Its Applications - Chapter 1 theorems and definitions well answered questions Linear Algebra

Theorems and definitions for Linear Algebra

and Its Applications - Chapter 1 correctly

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A system of linear equations has

1. no solution, or
2. exactly one solution, or
3. infinitely many solutions. A system of linear equations is said to be consistent if it has either one solution or infinitely many solutions; a system is inconsistent if it has no solution. Elementary Row Operations steps
4. (Replacement) Replace one row by the sum of itself and a multiple of another row.
5. (Interchange) Interchange two rows.
6. (Scaling) Multiply all entries in a row by a nonzero constant. TWO FUNDAMENTAL QUESTIONS ABOUT A LINEAR SYSTEM
7. Is the system consistent; that is, does at least one solution exist?
8. If a solution exists, is it the only one; that is, is the solution unique? A leading entry in a row refers to... the leftmost nonzero entry (in a nonzero row). A rectangular matrix is in echelon form (or row echelon form) if it has the following three properties:
9. All nonzero rows are above any rows of all zeros.
10. Each leading entry of a row is in a column to the right of the leading entry of the row above it.
11. All entries in a column below a leading entry are zeros. If a matrix in echelon form satisfies the following additional conditions, then it is in reduced echelon form (or reduced row echelon form):

Algebraic Properties of Rn

• Shows properties of scalars and how they interact with matrices. What is a Linear Combination A linear combination is the combination of all possible positions generated by the matrix and scalars (the grid lines align with the origin, a symmetric line transformation on a plane.) Span definition The collection of vectors that lay within the span are denoted by Span{v ... vp}, this is a subset. Scalars can also be applied in this form: c1v1 + c2v

• ... + CpVp. (c are scalars) The span The collection of all possible positions that collect on a Cartesian plane to make a sheet. (If all points are independent of each other.) Ax=b definition The linear combination of a matrix (A) and a vector (x) is denoted by: Ax=b

Ax=b Matrix - Theorem 3 The equation Ax = b has a solution if and only if b is a linear combination of the columns of A (can it be row reduced echelon form). Augmented matrix operations (RREF) is the same as AX=b operations Ax=b Matrix logical equivalent -Theorem 4 (All these thing must be true for the matrix to be valid.) Let A be an M x N matrix. Then the following statements are logically equivalent. That is, for a particular A, either they are all true statements or they are all false. a. For each b in Rm, the equation Ax D b 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. Row-Vector Rule for Computing Ax If the product Ax is defined, then the i-th entry in Ax is the sum of the products of corresponding entries from row i of A and from the vector x. (I-th row is the sum of the Ax=b elements, augmented by the vector.) Ax=b Matrix Scalar rules/steps -Theorem 5 What is the trivial solution?