# Linear Algebra - Basis of column space

• tg22542
In summary, the conversation discusses finding the basis and coordinates of the column space of a given matrix, as well as determining the rank and finding a basis for the row space. The solution involves using Gauss-Jordan operations to reduce the matrix and identify linearly independent columns. The coordinates of the dependent columns can be found using coordinate vectors.
tg22542

## Homework Statement

Let A be the matrix
A =
1 −3 −1 2
0 1 −4 1
1 −4 5 1
2 −5 −6 5

(a) Find basis of the column space. Find the coordinates of the dependent columns relative
to this basis.
(b) What is the rank of A?
(c) Use the calculations in part (a) to ﬁnd a basis for the row space.

---

## The Attempt at a Solution

I used Gauss-Jordan operations on the matrix to solve it down to :

1 0 -13 5
0 1 -4 1
0 0 1 0
0 0 0 0

From here we can see which columns are linearly independent and which are dependent. But I don't understand what they want me to write for a solution for the coordinates.

Would they simply be:

(1,0,-13)
(0,1,-4)
(0,0,1)

??

b) Not sure exactly what this means even after researching, how do I determine the rank ?

c) I feel I can do after I complete a)

Thanks

For part 1, by definition, ##Col(A) = span\{a_1, ... a_n\}## where ##a_1, ... a_n## are the linearly independent columns of ##A##.

The basis happens to be the set ##\{a_1, ... a_n\}## (without the "span" portion).

Also, do you know about coordinate vectors?

Your given matrix has 4 numbers in each column. That is each column is in R^4. So how can the span of {(1,0,-13), (0,1,-4), (0,0,1)} be subset of R^4?. You need to get your definitions done perfectly before you can solve these problems.

## 1. What is the column space in linear algebra?

The column space in linear algebra is the span of the columns of a matrix. It is the set of all possible linear combinations of the columns of a matrix. In other words, it is the space that is spanned by the columns of a matrix.

## 2. How is the basis of a column space determined?

The basis of a column space is determined by finding a set of linearly independent columns from the original matrix. These columns will form the basis for the column space, meaning that any other column in the space can be written as a linear combination of these basis columns.

## 3. What is the significance of the basis of a column space?

The basis of a column space is significant because it allows us to represent any column in the space as a linear combination of the basis columns. This makes it easier to solve systems of linear equations and perform other operations on the column space.

## 4. Can a column space have more than one basis?

Yes, a column space can have more than one basis. This is because there can be multiple sets of linearly independent columns in a matrix that can form a basis for the column space. However, all of these bases will have the same number of columns, known as the dimension of the column space.

## 5. How is a basis for the column space related to the rank of a matrix?

The number of basis columns for the column space is equal to the rank of the matrix. This means that the rank of a matrix can also be thought of as the dimension of the column space. In other words, the rank of a matrix tells us the maximum number of linearly independent columns in the matrix, which will form the basis for the column space.

• Calculus and Beyond Homework Help
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
15
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
567
• Calculus and Beyond Homework Help
Replies
0
Views
598
• Calculus and Beyond Homework Help
Replies
2
Views
787
• Calculus and Beyond Homework Help
Replies
3
Views
2K
• Calculus and Beyond Homework Help
Replies
18
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
10
Views
2K
• Calculus and Beyond Homework Help
Replies
2
Views
1K