Tiêu đề Vector Space_01.pdf ✅

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)