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 onetoone if and only of kerA = {0}.
2. A is onto if it has maximal rank, ranA = Cm .
3. If A ∈ Mn , then it is onetoone 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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 Fall '12
 BernhardBodmann
 Math, Matrices

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