Orthogonal subspaces
Not what you're looking for?
Let A be an mxn matrix. show that
1) If x Є N(A^TA), then Ax is in both R(A) and N(A^T).
2) N(A^TA) = N(A.)
3) A and A^TA have the same rank.
4) If A has linearly independent columns, then A^TA is nonsingular.
Let A be an mxn matrix, B an nxr matrix, and C=AB. Show that:
1) N(B) is a subspace of N(C).
2) N(C) perp. is a subspace of N(B) perp. and consequently, R(C^T) is a subspace of R(B^T).
Purchase this Solution
Solution Summary
There are several proofs here regarding matrices, including proving same rank, nonsingularity, and subspaces.
Purchase this Solution
Free BrainMass Quizzes
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.
Geometry - Real Life Application Problems
Understanding of how geometry applies to in real-world contexts
Probability Quiz
Some questions on probability
Solving quadratic inequalities
This quiz test you on how well you are familiar with solving quadratic inequalities.