Fundamental subspace theorem
Not what you're looking for?
Consider the matrix
a=(1 1 2
1 2 -1
3 2 1
5 5 2)
Find N(A), R(A), N(A^T),R(A^T). Show that the fundamental subspace theorem holds: N(A^T)=R(A)^(upside down T), N(A)=R(A^T)^(upsidedown T).
Hint: Notice that the fourth row is the sum of the first three rows.
Purchase this Solution
Solution Summary
This gives a matrix and then proves that the fundamental subspace theorem holds. Matrix algebra is analyzed.
Solution Preview
The solution is attached.
Problem:
a=(1 1 2
1 2 -1
3 2 1
5 5 2)
Find N(A), R(A), N(A^T),R(A^T). Show that the fundamental subspace theorm holds: N(A^T)=R(A)^(upside down T), N(A)=R(A^T)^(upsidedown T).
Hint: Notice that the fourth row is the sum of the first three rows.
Solution:
Let us apply the Gaussian Elimination to this matrix. Then, we get O1 A=U, OA=R, where
O1 =[1 0 0 0
-1 1 0 0
-4 1 1 0
-1 -1 -1 1 ]
U=[ 1 1 2
0 1 -3
0 0 -8
0 0 0]
O=[ -1/2 -3/8 5/8 0
1/2 5/8 -3/8 0
1/2 -1/8 -1/8 0
-1 -1 -1 1]
R=[ 1 0 0
0 1 0
0 0 1
0 0 0]
To get Matrix U and O1, we apply the Gaussian Elimination to the matrix A I
A I = [ 1 1 2 1 0 0 0
1 2 -1 0 1 0 0
3 2 1 0 0 1 0
5 5 2 0 0 0 1 ]
-- [1 1 2 1 0 0 0
0 1 -3 -1 1 0 0
0 -1 -5 -3 0 1 0
0 0 -8 -5 0 0 1 ...
Purchase this Solution
Free BrainMass Quizzes
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.