# Dimension of row/ column space

• jeffreylze
In summary, the conversation discusses how to verify that the row rank is equal to the column rank of a given matrix by showing that the rows are not linear combinations of each other and that one column is a linear combination of the other two columns. This explicitly proves that the rank of the matrix is equal to the row rank and column rank.
jeffreylze

## Homework Statement

In the following exercises verify that the row rank is equal to the column rank by explicitly finding the dimensions of the row space and the column space of the given matrix.

A = [1 2 1 ; 2 1 -1]

## The Attempt at a Solution

All i can think of is just row reduce it to row echelon form and then find the rank of the matrix. How do i do it explicitly?

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)

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)

Do you mean by one is NOT the multiple of the other?

You are quite correct. I did not proofread before submitting.

Okay, thanks. Now that make sense =D

## What is the dimension of row space?

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.

## How is the dimension of row space related to the number of columns in a 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.

## What is the difference between the dimension of row space and column 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.

## How is the dimension of row space related to the rank of a matrix?

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.

## Why is the dimension of row space important in matrix operations?

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.

• Linear and Abstract Algebra
Replies
4
Views
1K
• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Linear and Abstract Algebra
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
2K
• Calculus and Beyond Homework Help
Replies
4
Views
2K
• Calculus and Beyond Homework Help
Replies
8
Views
1K
• Calculus and Beyond Homework Help
Replies
10
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
10
Views
1K