Let A be an m ×n matrix of rank r>0 and let U be an echelon form of A . Explain why there exists

Let A be an m ×n matrix of rank r>0 and let U be an echelon...

Key Concepts

Elementary Row Operations

Elementary row operations are the fundamental procedures (such as row swaps, scaling, and adding multiples of one row to another) used to simplify a matrix to a form that is easier to work with, such as echelon form. These operations preserve the solution set of the associated system of linear equations and can be represented as multiplication by invertible matrices. This key concept underpins the idea that transforming a matrix into echelon form is a reversible process.

Invertible Matrices

An invertible matrix, often called a nonsingular matrix, is one that has a multiplicative inverse such that when it multiplies with its inverse, the result is the identity matrix. In the context of transforming a matrix into echelon form, the matrix representing the sequence of elementary row operations is invertible. This guarantees that the transformation process is reversible, allowing us to express the original matrix as the product of an invertible matrix and its echelon form.

Echelon Form

Echelon form is a simplified version of a matrix obtained by performing elementary row operations. In this form, all rows consisting entirely of zeros are at the bottom, and each leading (nonzero) entry of a row is to the right of the leading entry of the previous row. This structured form helps in easily identifying the rank of the matrix and in solving linear systems, as the position and number of nonzero rows directly relate to the matrix's row linear independence.

Matrix Rank

The rank of a matrix is defined as the maximum number of linearly independent rows or columns in the matrix. It provides a measure of the dimensionality of the row or column space. In the discussion of echelon forms, the number of nonzero rows corresponds exactly to the rank, offering a clear view of the matrix’s essential structure, which is crucial for both solving linear algebra problems and for performing further decompositions.

Rank-1 Matrix Decomposition

A rank-1 matrix is a matrix that can be written as the outer product of two nonzero vectors. Any matrix of rank r can be decomposed into a sum of r rank-1 matrices, each representing a fundamental component of the original matrix. This decomposition is significant in understanding the structure of the matrix, as it breaks down the complex transformations encoded in the matrix into simpler, more digestible parts that capture the essence of the transformation in a basic form.

Transcript

00:01 All right, problem 34, we have a matrix that is m by n and has rank positive and the action form of a.

00:24 So the first thing we need to do is to find an invertible matrix such that a equals e times u.

00:31 So recall, because u is the is the each in form of a, recall how we do the row reduction, and that is the same as we times a by some matrix e1 to get a new matrix, say u1, because the row reduction is a linear transformation and the rule reduction is clearly invertible.

01:09 So each of this e1 elementary matrix will be invertible.

01:17 And actually, usually the only once of the rule reduction will not be enough.

01:24 So we have plenty of a bunch of road reductions.

01:27 So that gives us a bunch of e1.

01:30 So that is, so i'll write it here.

01:36 So that means we have, say we have p times, p times.

01:43 We have p times rule reduction, so that's ep times until e1.

01:53 The operation of a gives u1, sorry, just u.

02:00 And since each of the elementary matrix is invertible, so the whole matrix is invertible.

02:08 So that means a is epp till e1 inverse times u.

02:19 So we just need to take e to be e p to e1 inverse.

02:34 So we're done.