Describe the possible echelon forms for matrices with the fo

Describe the possible echelon forms for matrices with the following properties: (a) A is a 2 times 2 matrix with linearly dependent columns. (b) A is a 4 times 3 matrix, A = [a_1 a_2 a_3], such that {a_1, a_2} is linearly independent and a_3 is not in the span of a_1 and a_2.

Solution

A is an 2*2matrix with linear dependent columns

n this section, we describe a method for finding the rank of any matrix. This method assumes familiarity with echelon matrices and echelon transformations.

The maximum number of linearly independent vectors in a matrix is equal to the number of non-zero rows in its row echelon matrix. Therefore, to find the rank of a matrix, we simply transform the matrix to its row echelon form and count the number of non-zero rows.

Consider matrix A and its row echelon matrix, A_ref. Previously, we showed how to find the row echelon form for matrix A.

Because the row echelon form A_ref has two non-zero rows, we know that matrix A has two independent row vectors; and we know that the rank of matrix A is 2.

You can verify that this is correct. Row 1 and Row 2 of matrix A are linearly independent. However, Row 3 is a linear combination of Rows 1 and 2. Specifically, Row 3 = 3*( Row 1 ) + 2*( Row 2). Therefore, matrix A has only two independent row vectors

0 1 2 1 2 1 2 7 8		1 2 1 0 1 2 0 0 0
A		Aref