MATH 21 Lecture Notes - Lecture 11: Row Echelon Form, Transpose, Gaussian Elimination

37 views3 pages

orchidpig108

20 Jul 2018

School

Department

Course

Professor

For unlimited access to Class Notes, a Class+ subscription is required.

Document Summary

If there exist (cid:1866) (cid:1866) matrix (cid:1828) such that (cid:1827)(cid:1828)=(cid:1828)(cid:1827)=(cid:3041), (cid:1828)= inverse of (cid:1827) Or: nonsingular, no inverse = non-invertible or singular. [(cid:1827),(cid:1828),(cid:1829)=(cid:1866) (cid:1866) matrices (cid:1828)(cid:1827)=(cid:1827)(cid:1829)=(cid:3041) (cid:1828)=(cid:1829)] if (cid:1866) (cid:1866) matrix (cid:1827) has an inverse, matrix (cid:1827) is unique. (cid:1827) is invertible (cid:1866) (cid:1866) matrix; there is only one solution to matrix equation and sequence is: assume (cid:1827) and (cid:1828) are invertible. Then (cid:1827)(cid:1828) is invertible and (cid:4666)(cid:1827)(cid:1828)(cid:4667) 1=(cid:1828) 1(cid:1827) 1: if (cid:1827) is invertible then (cid:1827) 1 is invertible and (cid:4666)(cid:1827) 1(cid:4667) 1=(cid:1827) Theorem: assume (cid:1827) is invertible: if (cid:1828),(cid:1829) are matrices and (cid:1827)(cid:1828)=(cid:1827)(cid:1829) then (cid:1828)=(cid:1829, if (cid:1828),(cid:1829) are matrices and (cid:1827)(cid:1828)=(cid:1829)(cid:1827) then (cid:1828)=(cid:1829) Let (cid:1827)=(cid:4672)(cid:1853) (cid:1854) (cid:1855) (cid:1856)(cid:4673) a is invertible if and only if determinant (cid:4666)(cid:1853)(cid:1856) (cid:1854)(cid:1855)(cid:4667) (cid:882) (cid:883)(cid:1853)(cid:1856) (cid:1854)(cid:1855)(cid:4672)(cid:1856) (cid:1854) (cid:1827) 1= (cid:1853)(cid:4673) 1 (cid:3031)(cid:3032)(cid:3032)(cid:3040)(cid:3041)(cid:3041) gets inverse. ((cid:883) (cid:884) (cid:883) (cid:883) (cid:883) (cid:882)) (cid:884) (cid:887) (cid:886) 7 (cid:883) (cid:885) (cid:884: (cid:1827) is not invertible, make augment matrix, apply gaussian elimination.

Related Questions

Please show all work for credit and I only need questions number6, 27C, 28, and 33

Please help with all questions. Will rate ASAP.

This question addresses inverse transformations from a conceptual standpoint, a. Suppose that a linear transformation T: x rightarrow Ax from A to A takes some nonzero vector x to 0. Can this transformation be reversed? Does A exist? Suppose that a linear transformation T: x rightarrow Ax from is not one-to-one. What does that tell you about the set of vectors b that can be produced by applying the transformation T to all vectors x in A? How do you know, in that case, that the transformation T is not reversible (i.e. that there does not exist an inverse transformation with the standard matrix A-1 that can be applied to the result of Ax to restore the vector x to its original value)? Suppose that a linear transformation T: x rightarrow Ax from A to A is not onto. What does that tell you about the set of vectors b that can be produced by applying the transformation T to all vectors x in A? Is it possible for a linear transformation x rightarrow Ax from A to A to be one-to-one but not onto, or onto but not one-to-one? If so, please give examples of each case. If not, explain why not. Is it possible for an arbitrary linear transformation x rightarrow Ax from A to B to be one-to-one but not onto, or onto but not one-to-one? If so. please give examples of each case. If not, explain why not. For what values of {a. b. c, d. e.f. a. h. i.j } will the matrix A = invertible? what is the least amount of partitioning that needs to be applied to M; to make M1 and M2 conformable for block multiplication? What is the product M = M1, M2 that you obtain using block multiplication with these partitions? What is the product M = M1 M2. that you obtain by multiplying the two matrices without partitioning? For which three values of a is the matrix B = not invertible, and why If M1 and M2 arc partitioned matrices such that M1 For each matrix product shown below, draw in the partitions that would need to be added to make the multiplication possible: Describe in general how the sizes of each of the partitions(#rows, #columns) in a matrix product C = AB are defined by the sizes of the partitions in each of the matrices A and B when partitioned matrices are multiplied.

8. Determine if each of the following statements is true or false. Clearly indicate your answer for each part by stating if it is true or false no justification is required). Each part is worth 2 marks for a correct answer; -1 for an incorrect answer to discourage guessing) (a) If A and B are n x n matrices, then rank(BA) rank(A). (b) If {A, B, C} is a linearly independent set of n x m matrices, then {AT, BT,CT} is a linearly independent set of m x n matrices. (C) W = {A E Rnxn A is invertible } is a subspace of Rnxn, where Rnxn denotes the vector space of real n x n matrices. (d) If xi and X2 are orthogonal vectors in R3 then it is always possible to find a unique vector X3 such that {x1, X2, X3} is an orthogonal basis for R3. (e) Let W be a subspace of R" and let x ER". Then projw(x - projw(x)) = 0.

MATH 21 Lecture Notes - Lecture 11: Row Echelon Form, Transpose, Gaussian Elimination

Document Summary

Get access

Related textbook solutions

Calculus

Single Variable Calculus: Early Transcendentals

CALCULUS:EARLY TRANSCENDENTALS

Related Documents

MATH 21 Study Guide - Midterm Guide: Row Echelon Form, Identity Matrix, Brie

MATH 21 Study Guide - Midterm Guide: Free Variables And Bound Variables, Linear Combination, Row Echelon Form

MATH 21 Study Guide - Final Guide: Linear Combination, Augmented Matrix, Row And Column Spaces

Related Questions