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Applied Mathematics 1411A/B Chapter Notes - Chapter 4.8: Row Echelon Form, Row And Column Spaces

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scarlethound445
29 Nov 2018
School
Western University
Department
Applied Mathematics
Course
Applied Mathematics 1411A/B
Professor
Bogdan Tudose

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Document Summary

4. 8 rank, nullity, and the fundamental matrix spaces. The row space and the column space of a matrix a have the same dimension. Nullity: the dimension of the null space of a dim(row space of a)=dim(row space of r) dim(column space of a)=dim(column space of r) If is any row echelon form of : Rank: the common dimension of the row space and column space of a matrix. If is an matrix: the number of leading variables in the general solution of. The number of parameters in the general solution of. If is a consistent linear system of equations in unknowns, and if has rank , then the general solution of the system contains parameters.

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Related Questions

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For the matrix A =
1 2 −1 1
3 6 −2 1
−1 −2 3 −5
and the column vector b =
−5
−13
9
(a) Find the reduced row echelon form of the augmented matrix [A|b].
(b) State the rank of A. Is the column space of A a line, a plane, or all of R3? Justify
your answer.
(c) Find the null space of A. Is the null space of A zero, a line, a plane, or all of R3?
Justify your answer.
(d) Find the general solution to the system Ax = b.

4. (12 points) Let the matrix A and its reduced row echelon form R be given by the following. A= [ 1 -2 2 -3 1 -2 4 -1 4 1 -3] [1001 1] -4 0 0 1 0 1 1 7 -1 0 0 1 1 -1 | = R. (a) Find a basis for the column space of A. (b) Find a basis for the row space of A. (c) Find a basis for the null space of A. (d) Verify that the dimension of the column space of A plus the dimension of the null space of A is equal to the number of columns of A.