Linear transformations and their matrix representations from a college linear algebra course.
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- What is a linear transformation T: V -> W?
- A function where T(u + v) = T(u) + T(v) and T(cu) = cT(u) for all vectors u, v and scalar c
- State the additivity property of linear transformations.
- T(u + v) = T(u) + T(v) for all vectors u, v in the domain
- State the homogeneity (scalar multiplication) property.
- T(cu) = cT(u) for all vectors u and scalars c
- What is the standard matrix of a linear transformation T: R^n -> R^m?
- The m by n matrix whose columns are T(e1), T(e2), ..., T(en), where e1, ..., en are standard basis vectors
- What is the kernel of a linear transformation T?
- The set of all vectors v such that T(v) = 0, denoted ker(T)
- What is the image of a linear transformation T?
- The set of all vectors w such that w = T(v) for some v in the domain
- State the rank-nullity theorem.
- rank(T) + nullity(T) = dim(domain) for any linear transformation T
- What is the rank of a linear transformation T?
- The dimension of the image of T
- What is the nullity of a linear transformation T?
- The dimension of the kernel of T
- When is a linear transformation T injective (one-to-one)?
- When ker(T) contains only the zero vector, i.e., nullity(T) = 0
- When is a linear transformation T surjective (onto)?
- When the image of T equals the codomain, i.e., rank(T) = dim(codomain)
- When is a linear transformation T bijective?
- When it is both injective and surjective
- When does a linear transformation T have an inverse?
- When T is bijective; the inverse T^-1 is also linear and unique
- If T has standard matrix A, when does T have an inverse?
- When A is invertible; the matrix of T^-1 is A^-1
- What is the composition (S compose T)(v)?
- S(T(v)), applying T first then S