Linear Algebra - Subspaces Quiz

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