Linear Algebra Exam 2

All the defs for 2.3-4.2

Linear Combination

A sum of a set of vectors, each multiplied by a scalar

Subspace

A vector space, contained inside another vector space

The span of a list of vectors

the set of all linear combinations of the vectors.

Spanning set of vectors for a Subspace

set of vectors whose span is a subspace, or the actual subspace

Linearly dependent

A vector set is linearly dependent if some vector can be expressed as a linear combination of the others

Linearly Independent

if no vector in the set can be expressed as a linear combination of the other vectors in the set

Basis

a set of linearly independent vectors that span the entire space

Dimension of a Subspace

The number of vectors in any basis of that subspace

Null Space

the set of all vectors x that, when multiplied by A, result in the zero vector

Nullity of a matrix

the dimension of its null space, which is the number of free variables in the system of equations

Column Space

The span of the column vectors in a matrix

Rank of a matrx

the number of non-zero rows or columns of a non-zero matrix

Row Space

the vector space spanned by the row vectors

Linear Transformation

A mapping between two vector spaces that preserves vector addition and scalar multiplication.

Cofactor

the signed determinant of the submatrix obtained by removing the row and column of that entry.