4 1. Preliminaries The image or range space of T is range(T ) = {y ∈ W | y = T (x) for some x ∈ V}. The set ker(T ) is a subspace of V, and range(T ) is a subspace of W. The dimension of ker(T ) is called the nullity of T , and the dimension of range(T ) is called the rank of T . Theorem 1.2 (Rank plus nullity theorem). Let V be a finite-dimensional vector space, and let T : V → W be a linear transformation from V to W. Then dim(ker(T )) + dim(range(T )) = dim(V). The map T is one-to-one (injective) if and only if ker(T ) = {0}, and T is onto (surjective) if and only if range(T ) = W. For finite-dimensional V, the map T is bijective (one-to-one and onto) if and only if ker(T ) = 0 and dim(V) = dim(W). For a finite-dimensional vector space V, Theorem 1.2 tells us that a linear transfor- mation T : V → V is one-to-one if and only if it is onto in this case it is invertible. We say T is an isomorphism if it is a bijection, and we say V and W are isomorphic if there exists an isomorphism from V onto W. If T is an isomorphism, then the set S ⊆ V is linearly independent if and only if T (S) is linearly independent, and S spans V if and only if T (S) spans W. Consequently, when T is an isomorphism, B is a basis for V if and only if T (B) is a basis for W. Hence, two finite-dimensional vector spaces V and W over F are isomorphic if and only if they have the same dimension. Theorem 1.3. Let B = {b1,..., bn} be a basis for the vector space V over F. Then the map T (v) = [v]B is an isomorphism from V onto Fn. As a consequence of Theorem 1.3, any n-dimensional vector space over F can be regarded as Fn. 1.4. Matrices An m × n matrix A, over F, is a rectangular array of mn elements of F arranged in m rows (horizontal lines) and n columns (vertical lines). The entry in row i and column j is denoted aij. For two m × n matrices A and B and scalars x, y, the matrix xA + yB is the m × n matrix with xaij + ybij in position ij. The dot product of two vectors x and y in Fn is x · y = ∑n i=1 xiyi. The product of an m × r matrix A and an r × n matrix B is the m × n matrix C = AB in which the (i, j) entry is the dot product of row i of A and column j of B. If A is an m × n matrix and x ∈ Fn is regarded as a column vector, then Ax is a column vector of length m and the ith entry is the dot product of row i of A with x. Note that column j of the matrix product AB is then A times column j of B. Also, row i of AB is the product of row i of A with the matrix B. For a matrix A, when we write A = (A1 A2 · · · An), we mean that Aj is the jth column of A. Note that Ax = ∑n i=1 xiAi is a linear combination of the columns of A. For an n × k matrix B with columns B1,...,Bk, we have AB = A(B1 B2 · · · Bk) = (AB1 AB2 · · · ABk).

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