Prove or give a counterexample: If U is the row echelon form of A, then img U=img A.

Name: Problem 36, Chapter 2: Applied Linear Algebra (Undergraduate Texts in Mathematics)
Uploaded: 2023-10-05T17:15:18-08:00
Duration: 3 min 1 s
Channel: William Mead
Description: Prove or give a counterexample: If U is the row echelon form of A, then img U=img A.

step 1 Let's suppose that A is an invertible matrix. Let's get some matrix A such that there exists A i

Prove or give a counterexample: If U is the row echelon form...

Key Concepts

Preservation of Subspaces under Transformations

When a matrix is multiplied on the left by an invertible matrix, such as when converting it to row echelon form via elementary row operations, the row space is preserved while the column space is generally not. Understanding which subspaces remain invariant under certain transformations is crucial in analyzing how different forms of a matrix reflect its essential properties.

Elementary Row Operations

Elementary row operations include swapping rows, scaling a row by a nonzero constant, and adding a multiple of one row to another row. These operations are used to transform a matrix into row echelon form and are reversible. They preserve the row space of the matrix but can alter the column space, which means that properties dependent on columns, like the image, might not remain invariant.

Row Echelon Form

Row echelon form is a simplified form of a matrix achieved through a series of row operations. It is used to reveal the structure of the matrix by making the leading entries (pivots) clear. While the row echelon form simplifies many problems such as solving linear systems, it does not necessarily preserve all properties of the original matrix, particularly the column space.

Matrix Image (Column Space)

The matrix image, often called the column space, is the set of all linear combinations of the columns of a matrix. It represents the range of the corresponding linear transformation and is fundamental in understanding the solvability of systems of linear equations and the dimensions related to the matrix.

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