# Solve Linear Algebra Exam: Find a Basis of U & Orthogonal Complement

• Striker2
In summary, you missed one of the most important topics in linear algebra. You should talk to your instructor about this.
Striker2
I have my linear algebra exam coming up but I missed the class on bases. Can anyone show me how this is solved?

2. Consider the subspace U of R4 defined by
U = span{(−1, 1, 0, 2), (1, 0, 0, 1), (2,−1, 1,−1), (0, 1, 0, 3)}
• Find a basis of U.
• Find a basis of the orthogonal complement U.

Then you missed one of the most important topics in linear algebra. I recommend you go talk to your instructor about this.

In any case I am sure your textbook says that a "basis" for a vector space has three properties:

1: Its vectors span the space.
2: Its vectors are independent.
3: The number of vectors in the space is equal to the dimension of the space.
(Actually, the "theorem" is that any two bases for the same space contain the same number of vectors. And then we define that number to be the dimesnsion.)\

And, any two of these imply the third. That is, if you know a set of vectors spans the space and is independent, then it is a basis and the number of vectors in it is the dimension of the space.

If you have a set of independent vectors and the number of vectors in it is equal to the dimension of the space, then it is a basis and spans the space.

If you have a set of vectors that span the space and the number of vectors in it is equal to the dimension of the space, then it is a basis and the vectors are independent.

Here, you are given that this set of vectors spans U. IF they are independent, then they are a basis. If they are not, then some of the vectors can be written terms of the others. Drop those and you haven't lost anything. What remains is a basis.

The vectors are "independent" if the only way to have a(−1, 1, 0, 2)+ b(1, 0, 0, 1)+ c(2,−1, 1,−1)+ d(0, 1, 0, 3)= (0, 0, 0, 0) is to have a= b= c= d. That is the same as (-a+ b+ 2c, a- c+ d, c, 2a+b-c+ 3d)= (0, 0, 0, 0) which means, of course, -a+ b+ 2c= 0, a- c+ d= 0, c= 0,, 2a+b-c+ 3d= 0. Try solving those equations for a, b, c, and d. An obvious solution is a= b= c= d= 0. If that is the only solution, then the vectors given are independent and form a basis. If not, then at least one can be written in terms of the others. Say, perhaps, that c and d can be written interms of a and b. Then you don't need the vectors multiplied by c and d and just those multiplied by a and b are necessary. That would be your basis.

Awesome! Thanks for the help, I understand completely.

## What is a basis of a vector space?

A basis of a vector space is a set of linearly independent vectors that span the entire vector space. This means that any vector in the vector space can be written as a linear combination of the basis vectors.

## What is the purpose of finding a basis of U?

Finding a basis of U allows us to represent any vector in the subspace U in terms of a linear combination of the basis vectors. This makes it easier to perform calculations and understand the properties of the subspace.

## What is the definition of an orthogonal complement?

The orthogonal complement of a subspace U is the set of all vectors that are perpendicular to every vector in U. In other words, it is the set of all vectors that form a right angle with every vector in U.

## How is the orthogonal complement related to the basis of U?

The basis of U is a set of linearly independent vectors that span the subspace U. The orthogonal complement is a set of vectors that are perpendicular to every vector in U. This means that the basis of U and the basis of the orthogonal complement are two sets of vectors that together span the entire vector space.

## Why is it important to find the basis of U and the orthogonal complement?

Finding the basis of U and the orthogonal complement allows us to understand the properties of a vector space and its subspaces. It also helps with performing calculations and solving problems related to the vector space. Additionally, these concepts are important in many areas of mathematics and physics, such as linear algebra, quantum mechanics, and signal processing.

• Linear and Abstract Algebra
Replies
9
Views
1K
• Linear and Abstract Algebra
Replies
9
Views
744
• Linear and Abstract Algebra
Replies
2
Views
900
• Linear and Abstract Algebra
Replies
3
Views
2K
• Linear and Abstract Algebra
Replies
8
Views
1K
• Linear and Abstract Algebra
Replies
3
Views
2K
• Linear and Abstract Algebra
Replies
3
Views
2K
• Linear and Abstract Algebra
Replies
2
Views
1K
• Linear and Abstract Algebra
Replies
5
Views
1K
• Linear and Abstract Algebra
Replies
52
Views
2K