# Vector Space

**Topics:**Linear algebra, Matrices, Matrix

**Pages:**17 (4063 words)

**Published:**August 28, 2013

Let V be a non-empty set of certain objects. Assume two algebraic operations (i) vector addition (ii) scalar multiplication deﬁned on the elements of V .The set V is called a vector space if the following properties are satisﬁed ∀ u, v, w ∈ V and any scalars α, β.

Vector Space

Let V be a non-empty set of certain objects. Assume two algebraic operations (i) vector addition (ii) scalar multiplication deﬁned on the elements of V .The set V is called a vector space if the following properties are satisﬁed ∀ u, v, w ∈ V and any scalars α, β. Properties with respect to vector addition ‘+’ V is closed under vector addition: u+v ∈V Vector addition is commutative: u+v =v+u Vector addition is associative: (u + v) + w = u + (v + w) Vector addition has an identity element: ∃ e∈V :u+e=e+u=u Vector addition has inverse elements: ∃ u−1 ∈ V : u + u−1 = e = u−1 + u

Properties with respect to scalar multiplication

V is closed under scalar multiplication: αu ∈ V Distributivity holds for scalar multiplication over ﬁeld addition: (α + β)u = αu + βu Distributivity holds for scalar multiplication over vector addition: α(u + v) = αu + αv Scalar multiplication is compatible with multiplication in the ﬁeld of scalars: α(βu) = (αβ)u = β(αu) Scalar multiplication has an identity element: 1u = u = u1

Examples I

Check whether the following form a vector space?

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The set of real numbers with usual addition and multiplication. The set of integers with usual addition and multiplication. The set of positive integers with usual addition and multiplication. The set of all ordered n-tuples of real numbers with ‘+’ and ‘.’ deﬁnes as: If u = (x1 , x2 , ..., xn ), v = (y1 , y2 , ..., yn ), then u + v = (x1 + y1 , x2 + y2 , ..., xn + yn ) αu = (αx1 , αx2 , ..., αxn ) The set of all polynomials of degree less than or equal to n The set of all polynomials of degree equal to n

Vector Space

Vector Space

Linear Combination of vectors: Let {u1 , u2 , . . . , un } be n vectors and {α1 , α2 , . . . , αn } be n scalers. Linear Combination: α1 u1 + α2 u2 + . . . αn un

Vector Space

Linear Combination of vectors: Let {u1 , u2 , . . . , un } be n vectors and {α1 , α2 , . . . , αn } be n scalers. Linear Combination: α1 u1 + α2 u2 + . . . αn un Span of a ﬁnite set of vectors: Set of all possible linear combinations of the vectors of the set.

Vector Space

Linear Combination of vectors: Let {u1 , u2 , . . . , un } be n vectors and {α1 , α2 , . . . , αn } be n scalers. Linear Combination: α1 u1 + α2 u2 + . . . αn un Span of a ﬁnite set of vectors: Set of all possible linear combinations of the vectors of the set. Linear Dependence, Independence: a family of vectors is linearly independent if none of them can be written as a linear combination of ﬁnitely many other vectors in the collection. If α1 u1 + α2 u2 + . . . αn un = 0 =⇒ α1 = α2 = . . . = αn = 0, then u1 , u2 , . . . , un are LI.

Vector Space

Linear Combination of vectors: Let {u1 , u2 , . . . , un } be n vectors and {α1 , α2 , . . . , αn } be n scalers. Linear Combination: α1 u1 + α2 u2 + . . . αn un Span of a ﬁnite set of vectors: Set of all possible linear combinations of the vectors of the set. Linear Dependence, Independence: a family of vectors is linearly independent if none of them can be written as a linear combination of ﬁnitely many other vectors in the collection. If α1 u1 + α2 u2 + . . . αn un = 0 =⇒ α1 = α2 = . . . = αn = 0, then u1 , u2 , . . . , un are LI. Dimension and Basis: An LI set which spans the whole vector space is called basis. The number of elements in a basis is called the dimension of that vector space.

Important Results

Important Results

Let S be a non empty subset of a vector space V , then [S], the span of S, is a subspace of V .

Important Results

Let S be a non empty subset of a vector space V , then [S], the span of S, is a subspace of V . The set {v} is LD iﬀ v = 0

Important Results

Let S be a non...

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