Vector Spaces: Orthogonal Complements

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Questions and Answers

What does it mean for a vector to be orthogonal to a vector space V?

The vector v is orthogonal to every vector in V.

What is the orthogonal complement of a vector subspace W in vector space V?

W⊥ (W perp), which is the set of all vectors orthogonal to all vectors in W.

What is the orthogonal complement to the entire vector space V?

{0}

What is W∩W⊥, where W is a subspace of V?

{0} only Signup and view all the answers

What does the equation V = W + W⊥ represent?

It indicates that if V is a vector space and W is a subspace of that vector space, it is possible to express V in terms of W and its orthogonal complement. Signup and view all the answers

What is the implication of the Orthogonal Decomposition (Projection Theorem)?

Any vector in V can be written as a component in W and a component in W⊥, where W is a subspace of V. Signup and view all the answers

What is (W⊥)⊥?

(W⊥)⊥ = W Signup and view all the answers

For an m x n matrix, what is the null space of A orthogonal to?

The row space of A. Signup and view all the answers

What does the null space of Aᵀ represent in relation to the column space?

The orthogonal complement of the column space of A. Signup and view all the answers

What is the left null space?

The null space of Aᵀ, which is the orthogonal complement of the column space of A. Signup and view all the answers

Why is the orthogonal complement of the row space of A the null space of A?

For a vector x to be in the null space of A, its dot product with all of the rows of A must be zero. Signup and view all the answers

What is the sum of the dimension of W and W⊥ if W is a subspace of Rⁿ?

dim(W) + dim(W⊥) = n Signup and view all the answers

What is the union of the row space and null space of A?

All of Rⁿ Signup and view all the answers

If x is a solution to Ax = b, how can x be orthogonally decomposed?

Into a vector in the row space of A and a vector in the null space of A, i.e., x = xᵣ + xₙ. Signup and view all the answers

What are the four fundamental subspaces of an m x n matrix A?

null space ↔ row space; null space of Aᵀ ↔ column space. Signup and view all the answers

What is the idea behind orthogonal projection?

The component of a vector that is parallel to another vector or subspace. Signup and view all the answers

What is the formula for an orthogonal projection?

Not provided in the content. Signup and view all the answers

How do you find the distance from a vector to a plane W?

||v - projᵥᵥv|| Signup and view all the answers

What is the vector in a subspace W that is closest to a vector v in V?

projᵥᵥv Signup and view all the answers

The orthogonal complement of a subset is always a ______.

subspace Signup and view all the answers

What should you do first when doing an orthogonal projection onto a multi-dimensional subspace?

Make sure to project onto an orthogonal basis of the subspace. Signup and view all the answers

How do you find the orthogonal complement for polynomials?

Set up a system of equations for the inner product being equal to zero and solve it. Signup and view all the answers

Study Notes

Orthogonal Basics

A vector is orthogonal to a vector space V if it is perpendicular to every vector in V.
Orthogonal complement (W⊥) of a vector subspace W in vector space V includes all vectors orthogonal to every vector in W.

Complement Properties

The orthogonal complement of the entire vector space V is represented by {0}, while the orthogonal complement of {0} is the entire vector space V.
The intersection of a subspace W and its orthogonal complement W⊥ is solely the zero vector {0}.

Vector Decomposition

In a vector space V, which includes subspace W, the relationship V = W + W⊥ holds true.
Any vector v in V can be expressed as a sum of components from W and W⊥.

Orthogonal Relationships

The orthogonal complement of the orthogonal complement (W⊥)⊥ returns the original subspace W.
For an m x n matrix A, the null space of A serves as the orthogonal complement to its row space.

Advanced Matrix Concepts

The null space of Aᵀ is the orthogonal complement of A's column space.
The left null space corresponds to the null space of Aᵀ, orthogonally complementing the column space of A.

Null Space and Row Space Dynamics

For a vector x to belong to A's null space, it must produce zero when dotted with all rows of A, confirming it's orthogonal to those rows.
The sum of the dimensions of W and W⊥ equals n, extending the idea of rank-nullity theorem.

Relationship of Subspaces

The union of the row space and null space of A encompasses all of Rⁿ.
If a vector x solves Ax = b, it can be decomposed into parts from the row space and null space, leading to the shortest vector in the row space achieving Ax = b.

Fundamental Subspaces

The four fundamental subspaces associated with an m x n matrix A include the null space, row space, null space of Aᵀ, and column space, revealing their orthogonal complement relationships.

Projection Insights

The concept of orthogonal projection determines the portion of a vector aligned with a subspace.
The formula for orthogonal projection helps identify components of vectors parallel to respective subspaces.

Distance and Proximity

The distance from a vector to a plane W is calculated through ||v - projᵥᵥv||, isolating the non-parallel component of the vector.
The vector in subspace W closest to vector v is identified as projᵥᵥv.

Subspace Complements

The orthogonal complement of any subset is necessarily a subspace, as it is formed from the span of the original subset.
When projecting onto a multi-dimensional subspace, ensure to project using an orthogonal basis to avoid redundancy.

Polynomial Orthogonality

For polynomial subspaces, derive the orthogonal complement by setting up and solving equations that satisfy the defined inner product being equal to zero.

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Description

Explore the concept of orthogonal complements in vector spaces through flashcards. This quiz covers definitions and properties related to orthogonality in vector subspaces, providing clarity on key terms. Perfect for students studying linear algebra.