(a) Write down a matrix of rank r whose first r rows do not form a basis for its row space. (b)

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(a) Write down a matrix of rank r whose first r rows do not...

Key Concepts

Elementary Row Operations

Elementary row operations include row swapping, row scaling, and row addition. These operations are used to manipulate matrices, particularly to obtain row echelon form without altering the row space, and understand how different orders of rows can affect the representation of the basis of the row space.

Row Echelon Form

Row echelon form is a simplified form of a matrix achieved through a series of elementary row operations, where every nonzero row begins with a leading entry (pivot) that is to the right of the pivot in the row above, and all rows consisting entirely of zeros are at the bottom of the matrix. This form is instrumental in solving linear systems and in identifying linearly independent rows.

Matrix Rank

Matrix rank refers to the number of linearly independent rows (or columns) in a matrix. It provides a measure of the matrix's non-degeneracy in representing linear transformations, determining how many dimensions the row space or column space spans, and is critical in solving systems of linear equations and in understanding the structure of the matrix.

Basis of the Row Space

A basis for the row space of a matrix is a set of linearly independent rows that span the entire row space. In many contexts, the nonzero rows obtained after transforming a matrix to row echelon form provide such a basis; however, simply taking the first r rows of a matrix of rank r may not suffice if those rows are not linearly independent or do not span the row space.

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