Linear Algebra Unit 4

Subspace

A subspace is a subset of a vector space that is also a vector space under the same operations of addition and scalar multiplication. It must contain the zero vector, be closed under vector addition, and be closed under scalar multiplication.

Kernel

The kernel of a linear transformation is the set of all vectors in the domain that are mapped to the zero vector in the codomain. It determines the degree of non-injectivity of the transformation.

Image

The image of a linear transformation is the set of all output vectors that can be obtained by applying the transformation to all vectors in the domain. It represents all possible resulting vectors in the codomain.

Basis 

A basis is a set of vectors in a vector space that is linearly independent and spans the space, meaning any vector in the space can be expressed as a linear combination of the basis vectors.

Coordinate Vector With Respect to some Basis

A coordinate vector with respect to a basis is a vector that expresses an original vector as a linear combination of the basis vectors. This set of coefficients characterizes the position of the original vector in the vector space defined by that basis.

Isomorphism

An invertible linear transformation.

• iff ker(T)= {0}, meaning the kernel of the transformation is trivial, and dim(im(T)) = dim(target space).

• If V is isomorphic to W then dim V = dim W

• Suppose T is a linear transformation V → W and ker(T) = {0}. If dimV=dimW, then T is an isomorphism.

• Suppose T is a linear transformation from V → W and im(T) = W. If dim V= dim W then T is an isomorphism.

Linear Transformation

A linear transformation is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. It can be represented by a matrix, and crucially, it satisfies the properties of additivity and homogeneity.

Matrix of a Transformation

The matrix of a transformation is a representation of a linear transformation with respect to given bases of the domain and codomain vector spaces. It encodes how the transformation acts on coordinate vectors by multiplying them with this matrix.