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Let v be any element of a vector space V, and let c be any scalar. ※ Each element in a vector space is called a vector, so a vector can be a real number, an n-tuple, a matrix, a polynomial, a continuous function, etc.ġ4 Theorem 2: Properties of scalar multiplication (the set of all real polynomials of degree n or less) ※ By the fact that the set of real numbers is closed under addition and multiplication, it is straightforward to show that Pn satisfies the ten axioms and thus is a vector space (4) Continuous function space : (the set of all real-valued continuous functions defined on the entire real line) ※ By the fact that the sum of two continuous function is continuous and the product of a scalar and a continuous function is still a continuous function, is a vector space (It is straightforward to show that these vector spaces satisfy the above ten axioms) (1) n-tuple space: Rn (standard vector addition) (standard scalar multiplication for vectors) (2) Matrix space : (the set of all m×n matrices with real-number entries) Ex: (m = n = 2) (standard matrix addition) (standard scalar multiplication for matrices)ġ2 (3) n-th degree or less polynomial space : (7) (8) (9) (10) ※ Any set V that satisfies these ten properties (or axioms) is called a vector space, and the objects in the set are called vectors ※ Thus, we can conclude that Rn is of course a vector spaceġ1 (the set of all m×n matrices with real-number entries)įour examples of vector spaces are introduced as follows. If the following ten axioms are satisfied for every u, v, and w in V and every scalar (real number) c and d, then V is called a vector space Addition: (1) u+v is in V (2) u+v=v+u (3) u+(v+w)=(u+v)+w (4) V has a zero vector 0 such that for every u in V, u+0=u (5) For every u in V, there is a vector in V denoted by –u such that u+(–u)=0 Let V be a set on which two operations (vector addition and scalar multiplication) are defined. Vector addition Scalar multiplication Treated as 1×n row matrix Treated as n×1 column matrix
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Let u, v, and w be vectors in Rn, and let c and d be scalars (1) u+v is a vector in Rn (closure under vector addition) (2) u+v = v+u (commutative property of vector addition) (3) (u+v)+w = u+(v+w) (associative property of vector addition) (4) u+0 = u (additive identity property) (5) u+(–u) = 0 (additive inverse property) (6) cu is a vector in Rn (closure under scalar multiplication) (7) c(u+v) = cu+cv (distributive property of scalar multiplication over vector addition) (8) (c+d)u = cu+du (distributive property of scalar multiplication over real-number addition) (9) c(du) = (cd)u (associative property of multiplication) (10) 1(u) = u (multiplicative identity property)ħ Notes: A vector in can be viewed as: a 1×n row matrix (row vector): or a n×1 column matrix (column vector): Rn-space : the set of all ordered n-tuples R1-space = set of all real numbers n = 1 (R1-space can be represented geometrically by the x-axis) n = 2 R2-space = set of all ordered pair of real numbers (R2-space can be represented geometrically by the xy-plane) n = 3 R3-space = set of all ordered triple of real numbers (R3-space can be represented geometrically by the xyz-space) n = 4 R4-space = set of all ordered quadruple of real numbersģ Notes: (1) An n-tuple can be viewed as a point in Rn with the xi’s as its coordinates (2) An n-tuple also can be viewed as a vector in Rn with the xi’s as its components Ex: or a point a vectorĤ (two vectors in Rn) Equality: if and only if Vector addition (the sum of u and v): Scalar multiplication (the scalar multiple of u by c): Notes: The sum of two vectors and the scalar multiple of a vector in Rn are called the standard operations in RnĦ Theorem 1: Properties of vector addition and scalar multiplication III: Mathematics (Paper II) 1 Vectors in Rnģ Subspaces of Vector Spaces 4 Spanning Sets and Linear Independence 5 Basis and Dimension Baljeet Singh, Assistant Professor, Post Graduate Government College, Sector 11, ChandigarhĢ 1 Vectors in Rn An ordered n-tuple : a sequence of n real numbers
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III: Mathematics (Paper II) 1 Vectors in Rn"- Presentation transcript:ġ Vector Spaces B.A./B.Sc. Presentation on theme: "Vector Spaces B.A./B.Sc.