# Linear Algebra - dimension of orthogonal complement

• GuiltySparks
In summary, the dimension of orthogonal complement in linear algebra is the number of linearly independent vectors that make up the subspace that is orthogonal to a given subspace. It is calculated using the formula dim(V) + dim(V⊥) = n, where n is the total number of dimensions in the vector space and V⊥ is the orthogonal complement of V. The dimension of orthogonal complement cannot be greater than the dimension of the original subspace and is at most n-1. This concept is closely related to linear independence and is important in decomposing vector spaces, solving linear equations, and understanding geometric properties. It can also provide a basis for the null space of a linear transformation, which has practical applications in fields such
GuiltySparks
I've attached a copy of the problem and my attempt at a solution.

This seems like a relatively straightforward question to me, but my answer seems to be the exact opposite of what the answer key says.

I reach the conclusion that the answer is C, but the answer is apparently D.

I'm beginning to think that the answer key is wrong. Can anybody verify my solution?

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## 1. What is the definition of the dimension of orthogonal complement in linear algebra?

The dimension of orthogonal complement in linear algebra refers to the number of linearly independent vectors that make up the subspace that is orthogonal to a given subspace. In other words, it is the number of dimensions needed to describe the space that is perpendicular to the original subspace.

## 2. How is the dimension of orthogonal complement calculated?

The dimension of orthogonal complement can be calculated using the formula dim(V) + dim(V) = n, where n is the total number of dimensions in the vector space and V is the orthogonal complement of V. This formula follows from the fact that the sum of the dimensions of two complementary subspaces must be equal to the total number of dimensions in the vector space.

## 3. Can the dimension of orthogonal complement be greater than the dimension of the original subspace?

No, the dimension of orthogonal complement cannot be greater than the dimension of the original subspace. This is because the orthogonal complement is a subspace that is perpendicular to the original subspace, and therefore must have fewer dimensions. In fact, the dimension of the orthogonal complement can be at most n-1, where n is the total number of dimensions in the vector space.

## 4. How is the dimension of orthogonal complement related to linear independence?

The dimension of orthogonal complement is closely related to linear independence. In fact, the vectors that make up the orthogonal complement are linearly independent, meaning that none of them can be written as a linear combination of the others. This is because they are perpendicular to each other, and therefore cannot be scaled to form one another.

## 5. Why is the concept of orthogonal complement important in linear algebra?

The concept of orthogonal complement is important in linear algebra because it allows us to decompose a vector space into two complementary subspaces. This can be useful in solving systems of linear equations and understanding the geometric properties of a vector space. Additionally, the orthogonal complement can provide a basis for the null space of a linear transformation, which has important applications in fields such as data analysis and machine learning.

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