Introduction Notes

# Rank and nullity have signicance for the map

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Unformatted text preview: to a basis of Cn . Next, show that rkA is the size of the complement. Rank and nullity have signiﬁcance for the map associated with a matrix A. 1. A is one-to-one if and only of kerA = {0}. 2. A is onto if it has maximal rank, ranA = Cm . 3. If A ∈ Mn , then it is one-to-one if and only if it is onto. In this case, A is called a bijection. 1.2 Dot product and orthogonality ￿ 1.2.3 Deﬁnition. For x, y ∈ Cm , we let ￿x, y ￿ = m xj yj , where xj and yj denotes the j th j =1 ￿ entry in the column vectors x or y , respectively. We also introduce the norm ￿x￿ = ￿x, x￿. We say that x and y are orthogonal, x ⊥ y , if ￿x, y ￿ = 0. If V is a subspace of Cm , then x ⊥ V means that x is orthogonal to all z ∈ V . We say that a set of vectors {z1 , z2 , . . . , zn } form an orthonormal system if ￿zj , zk ￿ = δj,k for all 1 ≤ j, k ≤ n. 1.2.4 Deﬁnition. Given an orthonormal system {z1 , z2 , . . . , zn }, we deﬁne the orthogonal projection onto the span V...
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