Minimal linearly dependent sets of columns
Not what you're looking for?
Given two matrices
{1 2 3 3 9
1 0 1 0 2
0 2 2 3 7
2 4 6 6 18}
{9 3 9 3
9 1 3 3
9 0 2 5
6 0 0 2
3 1 3 1}
for each one:
a) list all minimal linearly dependent sets of columns;
b) list all maximal linearly independent sets of columns;
c) list all minimal sets of columns which span all columns; a',b',c') The same for rows;
d) compute the rank.
Purchase this Solution
Solution Summary
The expert examines minimal linearly dependent sets of columns.
Solution Preview
Given two matrices
{1 2 3 3 9
1 0 1 0 2
0 2 2 3 7
2 4 6 6 18}
{9 3 9 3
9 1 3 3
9 0 2 5
6 0 0 2
3 1 3 1}
for each one:
a) list all minimal linearly dependent sets of columns;
b) list all maximal linearly independent sets of columns;
c) list all minimal sets of columns which span all columns;
a',b',c') The same for rows;
d) compute the rank.
Part 1. Let A=[├ █(1 2 3 3 9@1 0 1 0 2@0 2 2 3 7@2 4 6 6 18)]┤
Using elementary row operations, we obtain the reduced row echelon form of A
"rref"(A)=[├ █(1 0 1 0 2@0 1 1 □(3/2) □(7/2)@0 0 0 0 0@0 0 0 0 0)]┤
a)
Let's denote the column vectors of A by c_1,c_2,c_3,c_4,c_5 and the row vectors of A by r_1,r_2,r_3,r_4. Since "rref"(A) has leading 1's in the first and second column, it follows that {c_1,c_2} is a basis for the column space of A and {[1 0 1 0 2],[0 1 1 □(3/2) □(7/2)]} is a basis for the row space of A.
Since the column space of A has dimension 2, it follows that any set of 3 column vectors of A is linearly dependent. We see that no column of A is a scalar multiple of another column, so the minimal linearly dependent sets of A must all contain 3 column vectors of A. We see that there are (■(5@3))=10 possible combinations of 3 columns. So all minimal linearly dependent sets of columns of A are {c_1,c_2,c_3}, {c_1,c_2,c_4}, {c_1,c_2,c_5}, ...
Purchase this Solution
Free BrainMass Quizzes
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Probability Quiz
Some questions on probability
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.