Linear transformations and their matrix representations from a college linear algebra course.
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- What is a linear transformation T: V -> W?
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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.
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T(u + v) = T(u) + T(v) for all vectors u, v in the domain- State the homogeneity (scalar multiplication) property.
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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?
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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?
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The set of all vectors v such that T(v) = 0, denoted ker(T)- What is the image of a linear transformation T?
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The set of all vectors w such that w = T(v) for some v in the domain- State the rank-nullity theorem.
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rank(T) + nullity(T) = dim(domain) for any linear transformation T- What is the rank of a linear transformation T?
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The dimension of the image of T- What is the nullity of a linear transformation T?
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The dimension of the kernel of T- When is a linear transformation T injective (one-to-one)?
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When ker(T) contains only the zero vector, i.e., nullity(T) = 0- When is a linear transformation T surjective (onto)?
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When the image of T equals the codomain, i.e., rank(T) = dim(codomain)- When is a linear transformation T bijective?
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When it is both injective and surjective- When does a linear transformation T have an inverse?
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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?
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When A is invertible; the matrix of T^-1 is A^-1- What is the composition (S compose T)(v)?
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S(T(v)), applying T first then S
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