Linear Algebra - Exam 2

Vector space

a set V with two operations (vector addition and scalar multiplication) that satisfies the 10 vector space axioms

Subspace

a subset H of V that has 3 properties:

• the zero vector is in H

• H is closed under vector addition

• H is closed under multiplication by scalars

Column space (ColA)

the span of the columns of matrix A

Row space of A (RowA)

the span of the rows of A

Null space of A (NullA)

The set of all solutions to Ax = 0, the homogeneous equation

Kernel (null space) of a transformation T

Given a linear transformation T: V —> W between vector spaces, it is the set of all u in V such that T(u) = 0

Basis

A set of vectors that is linearly independent and spanning the vector space

Coordinates of x relative to a basis B

The weights used when expressing x as a linear combination of the basis vectors

Coordinate vector [x]_B

The column vector of coefficients used to represent x in a basis B

Dimension of a vector space

The number of vectors in any basis for the space

Cofactor expansion

Determinant of an nxn matrix can be computed by expanding along any row or column

Determinant of a triangular matrix

If A is triangular, det(A) = product of diagonal entries

Basis theorem

If a vector space has a dimension P,

• any set of p linearly independent vectors is a basis

• any set of P spanning vectors is a basis

Rank theorem

For an mxn matrix A, rank(A) + nullity(A) = n

Infinite-dimensional vector space

A vector space that cannot be spanned by a finite set