# Find bases for the following subspace of F^5

• zodiacbrave
In summary, to find bases for the subspaces W1 and W2 in F^5, we need to consider the constraints a1 - a3 - a4 = 0 and a2 = a3 = a4 and a1 + a5 = 0. These constraints give us an idea of how a characteristic element of each subspace should look like. For example, in W2, an element can be written as a linear combination of (1, 0, 0, 0, -1) and (0, 1, 1, 1, 0).
zodiacbrave

## Homework Statement

Find bases for the following subspaces of F^5:

W1 = {(a1, a2, a3, a4, a5) E F^5 : a1 - a3 - a4 = 0}

and

W2 = {(a1, a2, a3, a4, a5) E F^5: a2 = a3 = a4 and a1 + a5 = 0}

2. The attempt at a solution

Well, I understand a basis is the maximum amount of vectors in a set that are linearly independent, or the smallest amount of L.I vectors that span a space. What is throwing me off is the constraints a1 - a3 - a4 = 0 and a2 = a3 = a4 and a1 + a5 = 0

Well, the constraints give you an idea of how a characteristic element of the subspace should look like. For example, in W2 you have an element (a1, a2, a2, a2, -a1) = a1(1, 0, 0, 0, -1) + a2(0, 1, 1, 1, 0). This should give you an idea.

## 1. What is a subspace in F^5?

A subspace in F^5 is a subset of the vector space F^5 that satisfies the properties of a vector space, such as closure under addition and scalar multiplication.

## 2. How do you find bases for a subspace in F^5?

To find bases for a subspace in F^5, you can use the method of Gaussian elimination or the method of finding linearly independent vectors. These methods involve solving systems of equations and identifying linearly independent vectors that span the subspace.

## 3. What is the dimension of a subspace in F^5?

The dimension of a subspace in F^5 is the number of linearly independent vectors that span the subspace. This can also be thought of as the number of basis vectors that are needed to represent any vector in the subspace.

## 4. Can a subspace in F^5 have multiple bases?

Yes, a subspace in F^5 can have multiple bases. In fact, any set of linearly independent vectors that span the subspace can be considered a basis for that subspace.

## 5. How can finding bases for a subspace in F^5 be useful?

Finding bases for a subspace in F^5 can be useful for solving systems of linear equations, finding solutions to differential equations, and in other areas of mathematics and science where vector spaces are used. It can also help in understanding the structure and properties of a given subspace.

• Engineering and Comp Sci Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
1K
• Calculus and Beyond Homework Help
Replies
10
Views
8K
• Calculus and Beyond Homework Help
Replies
1
Views
3K
• Calculus and Beyond Homework Help
Replies
5
Views
1K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
15
Views
3K
• Introductory Physics Homework Help
Replies
11
Views
914
• Calculus and Beyond Homework Help
Replies
5
Views
9K
• Mechanical Engineering
Replies
1
Views
1K