hokie1 said:1. Show that the rows of A are not linear combinations of each other, i.e. one is the multiple of the other.
2. Show that one column is the linear combination of the other 2 columns. Then show that the remaining 2 columns are not a multiple of the other.
then you've explicitly shown that the rank(A) = row rank (A) = column rank (A)
The dimension of row space is the number of linearly independent rows in a matrix. It is also known as the row rank of a matrix. The dimension of row space can range from 0 to the number of rows in the matrix.
The dimension of row space is always less than or equal to the number of columns in a matrix. This is because the number of columns in a matrix represents the maximum number of linearly independent columns that can exist, and therefore the maximum dimension of the row space.
The dimension of row space refers to the number of linearly independent rows in a matrix, while the dimension of column space refers to the number of linearly independent columns. In other words, the dimension of row space is the maximum number of rows that can be used to create a linearly independent set, and the dimension of column space is the maximum number of columns that can be used.
The dimension of row space is equal to the rank of a matrix. The rank of a matrix is the maximum number of linearly independent rows (or columns) in a matrix. Therefore, the dimension of row space is a measure of the rank of a matrix.
The dimension of row space is important because it determines the number of linearly independent rows (or columns) that can be used to create a basis for the space spanned by the rows (or columns) of a matrix. This is crucial in solving systems of linear equations and in understanding the properties and behavior of a matrix.