Let A be an m×n matrix. Explain why the following are true. Any vector x in Rn can be uniquely written as a sum y+z, where y ∈ N(A) and z ∈ R(AT). Any vector b ∈ Rm can be uniquely written as a sum u+v, where u ∈ N(AT) and v ∈ R(A).

Let A be an m×n matrix. Explain why the following are true. Any vector x in Rn can be uniquely written as a sum y+z, where y ∈ N(A) and z ∈ R(AT). Any vector b ∈ Rm can be uniquely written as a sum u+v, where u ∈ N(AT) and v ∈ R(A).

Linear Algebra: A Modern Introduction

4th Edition

ISBN:9781285463247

Author:David Poole

Publisher:David Poole

Chapter3: Matrices

Section3.5: Subspaces, Basis, Dimension, And Rank

Problem 63EQ

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Question

Let A be an m×n matrix. Explain why the following are true.

Any vector x in Rⁿ can be uniquely written as a sum y+z, where y ∈ N(A) and z ∈ R(A^T).
Any vector b ∈ R^m can be uniquely written as a sum u+v, where u ∈ N(A^T) and v ∈ R(A).

Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.

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