Linear Algebra - Subspaces Quiz

Study Notes

Basic Properties of Subspaces

A subspace H of R^n must contain the zero vector, be closed under vector addition, and closed under scalar multiplication.

Addition and Scaling Subspace Property

Subspaces require closure under addition and scalar multiplication.

Origin

Inclusion of the origin is essential for any subspace; lines must pass through the origin to span a subspace in R^n.

Column Space

The column space of a matrix A consists of all linear combinations of its columns, defined as Col(A).

Column Space Example Explanation

For a matrix A = [a1...an] with columns in R^m, the column space can be represented as Span {a1,..., an}. It is a subspace of R^m and only equals R^m when its columns span R^m.

Column Space Subspace

The column space of an m x n matrix is contained in R^m, aligning with the number of elements in each column.

Null Space

The null space Nul(A) of a matrix A contains all solutions to the homogeneous equation Ax = 0.

Theorem 12

Null space Nul(A) of an m x n matrix is a subspace of R^n, representing solutions of the system Ax = 0 for m equations and n unknowns.

Null Space Subspace

The Null Space of an m x n matrix is confined to R^n.

Null Space Process

To verify if a vector v is in Nul(A), compute Av; if it returns the zero vector, v is in the null space. This implicit definition contrasts with the explicit formulation of the column space.

Basis of a Subspace

A basis for a subspace H of R^n is a linearly independent set that spans H.

Theorem 13

Pivot columns of a matrix A represent a basis for its column space, without alteration to their original form.

Dimension of a Subspace

The dimension of a subspace is determined by the number of vectors comprising it; the zero space {0} is assigned a dimension of 0.

Rank

The rank of a matrix A is equivalent to the dimension of its column space, represented as Rank(A) = dim(Col(A)). It also equals the number of pivot columns.

Theorem 14

For a matrix A with n columns, the relationship rank(A) + dim(Nul(A)) = n holds, linking the rank and nullity of the matrix.

Theorem 15

In a p-dimensional subspace H of R^n, any linearly independent set of p elements automatically forms a basis for H, as does any set that spans H.

Invertible Matrix Theorem - Continued

An n x n matrix A is invertible if and only if:
- Its columns form a basis for R^n
- Col(A) equals R^n
- dim(Col(A)) equals n
- rank(A) equals n
- Null space Nul(A) is {0}
- dim(Nul(A)) equals 0

Description

Test your understanding of subspaces in linear algebra with this quiz. Explore the basic properties, addition, and scaling, essential for mastering the concepts of R^n. Perfect for anyone looking to deepen their knowledge in this fundamental topic.

Linear Algebra - Subspaces Quiz

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Questions and Answers

What are the basic properties of subspaces?

A subspace is closed under addition and scalar multiplication.

What must every subspace include?

What is the column space of a matrix A?

How is the column space Col(A) related to the span?

The column space of an m x n matrix can be considered in R^n.

What is the null space of a matrix A?

The null space of an m x n matrix is a subspace of R^m.

How do you test if a vector v is in Nul(A)?

What is a basis for a subspace H of R^n?

The pivot columns of a matrix A form a basis for the column space of A.

What does the rank of a matrix define?

If a matrix A has n columns, then rank(A) + dim(Nul(A)) equals n.

What can be said about any linearly independent set of exactly p elements in a p-dimensional subspace H?

If Nul(A) equals the zero vector, then dim(Nul(A)) equals 0.