Title Vector Space_01.pdf ✅

Vector Spaces Page | 1 Topics • Vectors in Rnn; • Vector Spaces: Vectors in Abstract/General Spaces; • Subspaces of Vector Spaces; Linear Combinations • Spanning Sets and Linear Independence, Span of a Set; • Basis and Dimension of a Vector Space, Dimension of Subspaces; • Row Space and Column Space of a Matrix; • Rank of a Matrix • Coordinates and Change of Basis
Vectors in Rnn Page | 2 Definition 9.1.1. For nn ∈ N, we define Rnn = ��xx1 xx2 ⋮ xxnn �: xxii ∈ R for ii = 1, 2, ... , nn�. For example, � 1 −2 � ∈ R2, � −3 4 1 � ∈ R3, ⎣ ⎢ ⎢ ⎢ ⎡ −2 4 0 12 −3⎦ ⎥ ⎥ ⎥ ⎤ ∈ R5. The objects in Rnn are usually called vectors, more specifically geometric vectors. In this lecture, we will generalize the concepts of vectors and explore vectors other than the geometric ones. In other words, not every vector looks like an nn-tuple. Definition 9.1.2. Let xx = � xx1 xx2 ⋮ xxnn � and yy = � yy1 yy2 ⋮ yynn � be two vectors in Rnn. We say xx = yy if and only if xxii = yyii for all ii = 1, 2, ... , nn The sum of two vectors in Rnn and the scalar multiple of a vector in Rnn are called the standard operations in Rnn and are defined as follows: Definition 9.1.3. Let xx = � xx1 xx2 ⋮ xxnn � and yy = � yy1 yy2 ⋮ yynn � be two vectors in Rnn and cc be a real number. Then the sum xx + yy and scalar multiple ccxx are defined as xx + yy = � xx1 + yy1 xx2 + yy2 ⋮ xxnn + yynn � , ccxx = � ccxx1 ccxx2 ⋮ ccxxnn �. (9.1.1) Clearly, the vectors xx + yy and ccxx are in Rnn.
Vectors in Rnn Page | 3 Definition 9.1.4. Zero Vector The element in Rnn 00 = � 0 0 ⋮ 0 � is called the zero vector in Rnn. For example, 00 = � 0 0 � ∈ R2, 00 = � 0 0 0 � ∈ R3. Observe that 00 ∈ R2 and 00 ∈ R3 are not the same zero vectors. Definition 9.1.5. If xx = � xx1 xx2 ⋮ xxnn � be a vector in Rnn, then the vector −xx is called the negative of xx defined as −xx = � −xx1 −xx2 ⋮ −xxnn �. For example, if xx = � −1 4 −3 2� ∈ R4, then −xx = � 1 −4 3 −2 �. Theorem 9.1.1. (Properties of vector addition and scalar multiplication in Rnn) Let xx, yy, and zz be any vectors in Rnn, and cc, dd ∈ R. Then 1. xx + yy ∈ Rnn Closure Property: Rnn is closed under addition 2. xx + yy = yy + xx. Commutative Property of Rnn: addition is commutative 3. (xx + yy) + zz = xx + (yy + zz). Associative Property of Rnn: addition is associative 4. xx + 00 = xx for all xx ∈ Rnn Existence of Additive Identity 5. xx + (−xx) = 00 for all xx ∈ Rnn Existence of Additive Inverse
Vectors in Rnn Page | 4 6. ccxx ∈ Rnn Closure Property: Rnn is closed under scalar multiplication 7. cc(xx + yy) = ccxx + ccyy Distributive Property: Scalar distributes over vectors in Rnn 8. (cc + dd)xx = ccxx + ddxx Distributive Property: vector distributes over scalars 9. (cc )xx = cc(ddxx) Associative Property of Scalars 10. 1xx = xx. Multiplicative Identity Proof. 1. The proof immediately follows from eq. (9.1.1). 2. If xx = � xx1 xx2 ⋮ xxnn � and yy = � yy1 yy2 ⋮ yynn � be two vectors in Rnn, then xx + yy = � xx1 + yy1 xx2 + yy2 ⋮ xxnn + yynn � = � yy1 + xx1 yy2 + xx2 ⋮ yynn + xxnn � = yy + xx since xxii + yyii = yyii + xxii for all ii = 1, 2, ... , nn. 3. If xx = � xx1 xx2 ⋮ xxnn � , yy = � yy1 yy2 ⋮ yynn �, and zz = � zz1 zz2 ⋮ zznn � be vectors in Rnn, then (xx + yy) + zz = � xx1 + yy1 xx2 + yy2 ⋮ xxnn + yynn � + � zz1 zz2 ⋮ zznn � = � xx1 + yy1 + zz1 xx2 + yy2 + zz2 ⋮ xxnn + yynn + zznn � = � xx1 xx2 ⋮ xxnn � + � yy1 + zz1 yy2 + zz2 ⋮ yynn + zznn � = xx + (yy + zz)