Vector Spaces PDF

## Vector Spaces **ii) A field has at least two elements** **Vector space** Let *F* be a field and *V* be a non-empty set. In *V*, we define the operations of addition and scalar multiplication *×* where *x* ∈ *V* and *c* ∈ *F*. Then the set *V* is called a vector space over the field *F* if the...

## Vector Spaces **ii) A field has at least two elements** **Vector space** Let *F* be a field and *V* be a non-empty set. In *V*, we define the operations of addition and scalar multiplication *×* where *x* ∈ *V* and *c* ∈ *F*. Then the set *V* is called a vector space over the field *F* if the following axioms are satisfied: 1. (*V*, +) is an Abelian group * (i) *c*(x+y) = *c*x + *c*y * (ii) *(c₁+c₂)*x = *c₁*x + *c₂*x * (iii) *(c₁c₂)*x = *c₁*(*c₂*x) * (iv) 1 * x* = *x* * (v) 0 * x* = 0 2. The vectors space over the field *F* is denoted by *V(F)*. The elements of *F* are called scalars and the elements of *V* are called vectors. **WORKED EXAMPLES** * **1**. The set of all ordered pairs (*x₁ , x₂*) of the elements of the field of real nos forms a vector space w.r.t addition and scalar multiplication defined as * *(c₁ + c₂)* (*x₁, *x₂*) = (*c₁* *x₁* + *c₂* *x₁*, *c₁* *x₂* + *c₂* *x₂*) **Soln:** Let *V*(R) = {(*x₁ , x₂*)/ *x₁* ∈ R}. Let *α*, *β*, *Γ* ∈ *V*(R) *x* = (*x₁ , x₂*), *β* = (*y₁ , y₂*) *Γ* = (*z₁ , z₂*). Let *c*, *c₁*, *c₂* ∈ *F*. I. (*V*, +) is an Abelian group * i) x + *β* = (*x₁ , x₂*) + (*y₁ , y₂*) * = (*x₁+ *y₁* , *x₂* + *y₂*) * ii) *x* + (*β* + *Γ*) = (*x*+*β*) + *Γ* * *LHS:* *x* + (*β* + *Γ*) = (*x₁ , x₂*) + (*y₁ + *z₁* , *y₂* + *z₂*) * = (*x₁+*y₁** + *z₁* , *x₂* +*y₂* + *z₂*) * *RHS:* (*x* + *β*) + *Γ* = (*x₁ + *y₁* , *x₂* + *y₂*) + (*z₁ , *z₂*) * = (*x₁ + *y₁* + *z₁* , *x₂* + *y₂* + *z₂*) * ∴ LHS = RHS * iii) There exists 0 = (*0 , 0*) ∈ *V* such that * (*0 , 0*) + (*x₁ , x₂*) = (*x₁ , x₂*) + (*0 , 0*) = (*x₁ , x₂*) * iv) ∀ *x* ∈ *V* such that (*x₁ , x₂*) ∈ *V*, there exists - *x* = (-*x₁* , -*x₂*) * (*x₁ , x₂*) + (-*x₁* , -*x₂*) = (-*x₁* , -*x₂*) + (*x₁ , x₂*) = (*0 , 0*) = 0 * v) *x* + *β* = (*x₁ , x₂*) + (*y₁ , y₂*) * = (*x₁ + *y₁* , *x₂* + *y₂*) * = (*y₁ + *x₁* , *y₂* + *x₂*) * = *β* + *x* * ∴ (*V*, +) is an Abelian group. * ii) *c*(*x*+*β*) = *c*(*x₁ + *y₁* , *x₂* + *y₂*) * = *c*(*x₁ + *y₁* , *c*(*x₂* + *y₂*)) * = (*c* *x₁* + *c* *y₁* , *c* *x₂* + *c* *y₂*) * = *c*(*x₁ , *x₂*) + *c*(*y₁ , *y₂*) * = *c* *x* + *c* *β* * iii) *(c₁c₂)* *x* = *(c₁c₂)* (*x₁ , *x₂*) * = (*(c₁c₂)* *x₁* , *(c₁c₂)* *x₂*) * = *c₁*(*c₂* *x₁* , *c₂* *x₂*) * = *c₁*(*c₂* *x₁* , *c₂* *x₂*) * = *c₁*(*c₂* *x* , *c₂* *x₂*) * iv) 1 * x* = 1 * (*x₁ , *x₂*) = (*1* *x₁* , *1* *x₂*) = (*x₁ , x₂*) = *x* * v) 0 * x* = 0 * (*x₁ , *x₂*) = (*0* *x₁* , *0* *x₂*) = (*0 , 0*) = 0 * All the axioms of vector space are satisfied. ∴ *V* forms a vector space. * **2**. The set of all ordered triples (*x₁, *y₁, *z₁*) over the field of real nos forms a vector space w.r.t addition and scalar multiplication defined as * *(c₁ + c₂)*(*x₁, *y₁, *z₁*) + *(c₁* *x₁, *c₁* *y₁, *c₁* *z₁*) + *(c₂* *x₁, *c₂* *y₁, *c₂* *z₁*) * **3**. The set of all ordered n tuples of the elements of the field *F* forms a vector space w.r.t addition and scalar multiplication defined as * (*x₁, *x₂, ..., *x_n*) + (*y₁, *y₂*, ..., *y_n*) = (*x₁ + *y₁*, *x₂ + *y₂*, ..., *x_n* + *y_n*) * *c* (*x₁, *x₂*, ..., *x_n*) = (*c* *x₁, *c* *x₂*, ..., *c* *x_n*) *Proof* is the same as in the previous examples. * **4**. The vector space of real nos ordered *n* tuples over the field of real nos is denoted by *V_n*(R) or *Rⁿ* which is called the n-dimensional space *In particular*, if *n* = 2, the vector space is *V₂*(R) which is the two dimensional space, and if *n* = 3, the vector space is *V₃*(R) which is the three dimensional space. * **5**. P.T. the set of all real valued continuous (differentiable, integrable) funcs of *x* defined in the interval (0, 1) is a vector space. *Let *V* be the set of all real valued continuous func. of *x* defined in (0, 1) *Let *f* , *g* ∈ *V* and *c* ∈ R. Then *(*f*+*g*)(*x*) = *f*(*x*) + *g*(*x*) *(*c*f)(*x*) = *c*(*f*(*x*)) * ∴ (*V*, +) is an Abelian group. * i) If *f* and *g* are continuous funcs, then we know that their sum *f* + *g* is also continuous. * ii) If *f*, *g*, *h* ∈ *V*, then * *f* + (*g* + *h*)(*x*) = *f*(*x*) + (*g*+*h*)(*x*) * = *f*(*x*) + (*g*(*x*) + *h*(*x*)) * = [ *f*(*x*) + *g*(*x*)] + *h*(*x*) * = (*f*+*g*)(*x*) + *h*(*x*) * = [(*f*+*g*) + *h* ](*x*) * ∴ *f* + (*g* + *h*) = (*f*+*g*) + *h* * iii) The function 0(*x*) = 0 is the identity * (*0*+*f*)(*x*) = 0(*x*) + *f*(*x*) = 0 + *f*(*x*) = *f*(*x*) * ∴ *0* + *f* = *f* * (*f*+*0*)(*x*) = *f*(*x*) + 0(*x*) = *f*(*x*) + 0 = *f*(*x*) * ∴ *f* + 0 = *f* * ∴ *0* + *f* = *f* + *0* = *f* * iv) (-*f*)(*x*) = -(*f*(*x*)) is the additive inverse of *f* * ∴ (-*f*)(*x*) + *f*(*x*) = (-*f*+ *f*)(*x*) = 0(*x*) = 0 * ∴ *f*(*x*) + (-*f*)(*x*) = (*f* + (-*f*))(x) = 0(*x*) = 0 * v) (*f*+*g*)(*x*) = *f*(*x*) + *g*(*x*) * = *g*(*x*) + *f*(*x*) * = (*g*+*f*)(*x*) * ∴ *f* + *g* = *g* + *f* * ∴(*V*, +) is an Abelian group. * i) *c*(*f*+*g*) = *c*f + *cg* * *c*(*f*+*g*)(*x*) = *c*((f*+*g*)(*x*)) * = *c* [ *f*(*x*) + *g*(*x*)] * = *c* *f*(*x*) + *c* *g*(*x*) * = (*c*f)(*x*) + (*cg*)(*x*) * = (*c*f + *cg*)(*x*) * ∴ *c*(*f*+*g*) = *c*f + *cg* * ii) *(c₁+c₂)*f = *c₁*f + *c₂*f * *(c₁+c₂)*f(*x*) = *(c₁+c₂)* *f*(*x*) * = *c₁* *f*(*x*) + *c₂* *f*(*x*) * = (*c₁*f + *c₂*f)(*x*) * ∴ *(c₁+c₂)*f = *c₁*f + *c₂*f * iii) *(c₁c₂)*f = *c₁*(*c₂*f) * *(c₁c₂)*f(*x*) = *(c₁c₂)* *f*(*x*) * = *c₁* (*c₂* *f*(*x*)) * = *c₁* (*c₂*f)(*x*) * ∴ *(c₁c₂)*f= c₁ * (*c₂*f) * iv) 1 * f* = *f* * (1 * f*)(*x*) = 1 * *f*(*x*) = *f*(*x*) = *f*(*x*) * ∴ 1 * f* = *f* * v) 0 * f* = 0 * (0 * f*)(*x*) = 0 * *f*(*x*) = 0 = 0 * ∴ 0 * f* = 0 * ∴ *V* is a vector space over the field of real numbers. * **6**. The set of all ordered *n*-tuples of complex nos forms a vector space over the field of complex numbers. This is denoted by *Cⁿ* *Soln:* Let *V* = { (*z₁* , *z₂* , ..., *z_n*) / *z₁* , *z₂* , ..., *z_n* ∈ *C*}. *Let *α* = (*x₁* , *x₂* , ..., *x_n*), *β* = (*y₁* , *y₂* , ..., *y_n*) *Γ* = (*z₁* , *z₂* , ..., *z_n*) ∈ *V* *Let *c*, *c₁*, *c₂* ∈ *C*. * *c* *α* = *c* (*x₁* , *x₂* , ..., *x_n*) = (*c* *x₁* , *c* *x₂* , ..., *c* *x_n*) where *c* ∈ *C* I. (*V*, +) is an Abelian group * i) *α* + *β* = (*x₁* + *y₁* , *x₂* + *y₂* ,… *x_n* + *y_n* ) ∈ *V* * ii) *α* + (*β* + *Γ*) = (*α* + *β*) + *Γ* which can be easily verified * iii) 0 = (*0 , 0 , 0 , ... , 0*) ∈ *V*, the additive identity * iv) ∀ *x* = (*x₁ , *x₂* , ..., *x_n*) ∈ *V* such that there exists - *x* = (-*x₁* , -*x₂* , ..., -*x_n*) ∈ *V* * *x* + (-*x*) = -*x* + *x* = (*0 , 0 , ... , 0*) = 0 * v) *α* + *β* = *β* + *α* II. *c* (*α* + *β*) = *c* (*x₁*+*y₁* , *x₂*+*y₂* , ..., *x_n*+ *y_n*) * = *c* (*x₁*+*y₁* , *c* (*x₂*+*y₂*) , ..., *c* (*x_n*+*y_n*)) * = (*c* *x₁* + *c* *y₁* , *c* *x₂* + *c* *y₂* , ..., *c* *x_n* + *c* *y_n*) * = (*c* *x₁* , *c* *x₂* , ... , *c* *x_n*) + (*c* *y₁* , *c* *y₂* , ... , *c* *y_n*) * = *c* (*x₁* , *x₂* , ... , *x_n*) + *c* (*y₁* , *y₂* , ... , *y_n*) * = *c* *α* + *c* *β* * iii) *(c₁+c₂)* *α* = *c₁* *α* + *c₂* *α* which can be verified easily * iv) *(c₁c₂)* *α* = *c₁* (*c₂* *α*) which can be verified easily * v) 1 * α* = 1 * (*x₁* , *x₂* , ... , *x_n*) = (*1* *x₁* , *1* *x₂* , ... , 1 * *x_n*) = (*x₁* , *x₂* , ... , *x_n*) * ∴ *V* is a vector space over the field of complex numbers. ## Subspaces A non-empty subset *W* of a vector space *V* is said to be a subspace of *V* over a field *F* if *W* is a vector space over *F* w.r.t (the same operations as in *V*. * Ex: The set *W* of all ordered triplets (*x₁ , x₂ , 0*) over the field of real nos is a subset of the vector space *V₃*(R) w.r.t addition and scalar multiplication. The set *W* of all ordered triplets of the form *c*(*x₁ , *x₂ , 0*) is a subset of *V* and *W* is a subspace of *V*. * NOTE: Every vector space has always two subspaces: *0* and *V*. These are called trivial subspaces. *0* and any subspace of *V* is called a non-trivial subspace of *V*. **Criteria for a subset to be a subspace** Theorem 1: A non-empty subset *W* of a vector space *V* is a subspace of *V* iff * i) *x* ∈ *W* , *β* ∈ *W* ⇒ *x* + *β* ∈ *W* * ii) *c* ∈ *F*, *x* ∈ *W* ⇒ *c* *x* ∈ *W* **Note:** Whenever we have to prove that *W* is a subspace of *V*, it is enough to verify that *1* *x* *W* is a non-empty subset of *V*, and * ii) *x*, *β* ∈ *W*, *c₁*, *c₂* ∈ *F*, *c₁* *x* + *c₂* *β* ∈ *W* Theorem 2: A non-empty subset *W* of a vector space *V* is a subspace of *V* iff * i) *x*, *β* ∈ *W*, *c₁*, *c₂* ∈ *F* ⇒ *c₁* *x* + *c₂* *β* ∈ *W* **Note:** Whenever we have to prove that *W* is a subspace of *V*, it is enough to verify that *1* *x* ∈ *W* is a non-empty subset of *V*, and * ii) *x*, *β* ∈ *W*, *c₁*, *c₂* ∈ *F*, *c₁* *x* + *c₂* *β* ∈ *W* Theorem 3: The intersection of any two subspaces of a vector space *V* over a field *F* is a subspace of *V*. * **1**. P.T. the subset *W* = {(*x₁,*x₂, *x₃*)/ *x₁* + *x₂* + *x₃* = 0} of the vector space *V₃*(R) is a subspace of *V₃*(R) * **Soln:** *W* is a non-empty subset of *V₃*(R) since (*0 , 0 , 0*) ∈ *W*, and *c₁*, *c₂* ∈ *R* * Let *x* = (*x₁ , *x₂ , *x₃*) such that *x₁* + *x₂* + *x₃* = 0 * Let *β* = (*y₁ , *y₂ , *y₃*) such that *y₁* + *y₂* + *y₃* = 0 * *c₁* *x* + *c₂* *β* = *c₁* (*x₁ , *x₂ , *x₃*) + *c₂* (*y₁ , *y₂ , *y₃*) * = (*c₁* *x₁* + *c₂* *y₁* , *c₁* *x₂* + *c₂* *y₂* , *c₁* *x₃*+ *c₂* *y₃*) * = (c₁*x₁* + *c₂* *y₁* , *c₁* *x₂* + *c₂* *y₂* , *c₁* *x₃* + *c₂* *y₃*) * = (*c₁*x₁* + *c₂* *y₁* , *c₁* *x₂* + *c₂* *y₂* , *c₁*x₁* + *c₂*y₁* + *c₁*x₂* + *c₂*y₂*) * = (*c₁*x₁* + *c₂* *y₁* , *c₁* *x₂* + *c₂* *y₂* , 0) * ∴ *c₁* *x* + *c₂* *β* ∈ *W* * ∴ *W* is a subspace of *V₃*(R) * **2**. P.T. the subset *W* = {(*x₁, *y₁, *z₁*)/ *x₁* -3*y₁* + 4*z₁* = 0} of the vector space *R³* is a subspace of *R³* * **Soln:** *W* is a non-empty subset of *R³* since at least one element (0 , 0 , 0) ∈ *W* such that 0-3*0* + 4 *0* = 0. Let *x*, *β* ∈ *W* and *c₁*, *c₂* ∈ *R* * Let *x* = (*x₁ , *y₁ , *z₁*) such that *x₁* - 3*y₁* + 4*z₁* = 0 * Let *β* = (*x₂ , *y₂ , *z₂*) such that *x₂* - 3*y₂* + 4*z₂* = 0 * *c₁* *x* + *c₂* *β* = *c₁* (*x₁ , *y₁ , *z₁*) *+ *c₂* (*x₂ , *y₂ , *z₂*) * = (*c₁* *x₁* + *c₂* *x₂, *c₁* *y₁* + *c₂* *y₂*, *c₁* *z₁* + *c₂* *z₂*) * and *c₁* *x₁* + *c₂* *x₂* - 3(*c₁* *y₁* + *c₂* *y₂*) + 4(*c₁* *z₁* + *c₂* *z₂) * = *c₁* (*x₁* - 3 *y₁* + 4 *z₁*) + *c₂* (*x₂* - 3 *y₂* + 4 *z₂*) * = *c₁* (0) + *c₂* (0) = 0 * ∴ *c₁* *x* + *c₂* *β* ∈ *W* * ∴ *W* is a subspace of *V₃*(R). * **3**. P.T. the subset *W* = {(*x₁ , *y₁ , *z₁*)/ *x₁* = *y₁* = *z₁*} is a subspace of *V₃*(R). * **Soln:** *W* is a non-empty subset of *V₃*(R). * Let *x* ∈ *W*, *x* = (*x₁ , *y₁ , *z₁*) such that *x₁* = *y₁* = *z₁* * Let *β* ∈ *W*, *β* = (*x₂ , *y₂ , *z₂*) such that *x₂* = *y₂* = *z₂* * *c₁* *x* + *c₂* *β* = *c₁* (*x₁ , *y₁ , *z₁*) + *c₂* (*x₂ , *y₂ , *z₂*) * = (*c₁* *x₁* + *c₂* *x₂* , *c₁* *y₁* + *c₂* *y₂* , *c₁* *z₁* + *c₂* *z₂*) * and *x₁* = *y₁* = *z₁* and *x₂* = *y₂* = *z₂* * ∴ *x₁* = *y₁* = *z₁* ∴ *c₁* *x₁* + *c₂* *x₂* = *c₁* *y₁* + *c₂* *y₂* = *c₁* *z₁* + *c₂* *z₂* * ∴ *c₁* *x* + *c₂* *β* ∈ *W* * ∴ *W* is a subspace of * V₃*(R) * **4**. If a vector space is the set of real valued continuous funcs. over the field of real nos. Then P.T. the set *W* of solutions of the D.E. 2 *d²y*/d*x*² - 9 *dy*/d*x* + 2 *y* = 0 is a subspace of *V*. * **Soln:** *W* = { *f* / 2 *d²y*/d*x*² - 9 *dy*/d*x* + 2 *y* = 0} * Clearly *y = 0* satisfies the given D.E. * ∴ *W* is non-empty. * Let *y₁*, *y₂* ∈ *W* and *c₁*, *c₂* ∈ *R*. Then we have to show that *c₁* *y₁* + *c₂* *y₂* satisfies the D.E. * Consider * 2 *d²(*c₁* *y₁* + *c₂* *y₂*)/d*x*² - 9 *d(*c₁* *y₁* + *c₂* *y₂*) /d*x* + 2(*c₁* *y₁* + *c₂* *y₂*) * = 2 *c₁* *d² *y₁*/d*x*² + 2*c₂* *d² *y₂*/d*x*² - 9 *c₁* *dy₁*/d*x* - 9*c₂* *dy₂*/d*x* + 2 *c₁* *y₁* + 2 *c₂* *y₂* * = *c₁* ( 2 *d² *y₁*/d*x*² - 9 *dy₁*/d*x* + 2 *y₁*) * + *c₂* (2 *d² *y₂*/d*x*² - 9 *dy₂*/d*x* + 2 *y₂*) * = *c₁* (0) + *c₂* (0) = 0 * ∴ *c₁* *y₁* + *c₂* *y₂* ∈ *W* * ∴ *W* is a subspace of *V*. * **5**. Verify whether *W* = {(*x₁ , *x₂ , *x₃*)/ *x₁*² + *x₂*² + *x₃*² ≤ 1 } is a subspace of * V₃*(R) iff *W* is a subspace of *V₁*(R) * **Soln:** Consider *x* = (1 , 0 , 0) , *β* = ( 0 , 0 , 1) * Clearly *x* ∈ *W* since 1² + 0² + 0² = 1 ≤ 1 * *β* ∈ *W* since 0² + 0² + 1² = 1 ≤ 1 * Now *x* + *β* = ( 1 , 0 , 0 ) + ( 0 , 0 , 1 ) = ( 1 , 0 , 1 ) * ∴ 1² + 0² + 1² = 2 which is not less than or equal to 1. * ∴ *x* + *β* ∉ *W* * i.e., *x*, *β* ∈ *W* but *x* + *β* ∉ *W*. ∴ *W* is not a subspace of *V*. ## Linear span of a set **Linear Span** Def: Let *V* be a vector space over the field *F* and *y₁*, *y₂*. ... *y_m* be any *m* vectors of *V*. Any vector of the form *c₁* *y₁* + *c₂* *y₂* + .... + *c_n* *y_n* where *(c₁ , c₂ , ... , c_n)* ∈ *F* is called a linear combination of the vectors *y₁* , *y₂*, ... *y_m*. Def: Let *V* be a vector space over the field *F* and *S* be any non-empty subset of *V*. Then the linear span of Sis the set of all the linear combination of any finite no of elements of *S* and is denoted by *L(S)*. * ∴ *L(S)* = { *c₁* *y₁* + *c₂* *y₂* + .... + *c_m* *y_m* : *c_i* ∈ *F*, *i* = 1, 2, ... , *m*}. **LINEAR INDEPENDENCE & DEPENDENCE** Def: A set {*}y₁* , *y₂* , ... , *y_n*} of vectors of a vector space *V* over a field *F* is said to be linearly independent if there exist scalars *c₁* , *c₂* , *c_n* such that * c₁* *y₁* + *c₂* *y₂* + ... + *c_n* *y_n* = 0 then *c₁* = 0, *c₂* = 0, ... *c_n* = 0. Def: A set {*}y₁* , *y₂* , ... , *y_n*} of vectors of a vector space *V* over a field *F* is said to be linearly dependent if it is not linearly independent i.e. (the set { *y₁* , *y₂* , ... *y_n*} of vectors over a field *F* is said to be linearly dependent if there exist scalars *c₁* , *c₂* , *c_n* not all zero such that * c₁* *y₁* + *c₂* *y₂* + ... + *c_n* *y_n* = 0. **Note:** The null set of is always taken as a linearly independent set Given theorem 3 and 4 on the next page. * **1**. Show the set *S* = { (1, 0, 0), (0, 1, 0), (0, 0, 1)} is linearly independent in *V₃*(R) * **Soln:** Let *x₁* = (1, 0, 0), *x₂* = (0, 1, 0), *x₃* = (0, 0, 1) * Consider *c₁ *x₁ + *c₂* *x₂* + *c₃* *x₃* = (0, 0, 0) * ⇒ (*c₁* , 0 , 0) + (0 , *c₂* , 0) + (0 , 0 , *c₃*) = (0 , 0 , 0) * ⇒ (*c₁* , *c₂* , *c₃*) = (0 , 0 , 0) * ∴ *c₁* = 0 , *c₂* = 0 , *c₃* = 0 * ∴ *S* = { *x₁* , *x₂* , *x₃*} is linearly independent. * **2**. Show that the set *S* = { (1, 1, 1), (2, 2, 0), (3, 0, 0)} is linearly independent in *V₃*(R) * **Soln:** Let *x₁* = (1, 1, 1), *x₂* = (2, 2, 0), *x₃* = (3, 0, 0) * Consider *c₁* *x₁* + *c₂* *x₂* + *c₃* *x₃* = (0, 0, 0) * ⇒ (*c₁* + *c₂* + *c₃*, *c₁* + *c₂*, *c₃*) = (0, 0, 0) * ∴ *c₁* + *c₂* + *c₃* = 0, *c₁* + *c₂* = 0, *c₃* = 0 * ∴ *c₁* + *c₂* = 0, *c₃* = 0 * ∴ *c₁* + 2*c₂* + 3*c₃* = 0, *c₁* + 2*c₂* = 0, *c₃* = 0 * ∴ *c₁* = 0, *c₂* = 0, *c₃* = 0 * ∴ *S* = { *x₁* , *x₂* , *x₃*} is linearly independent. * **3**. P.T. the set *S₂* = { (1, 3, 2), (1, -7, -8), (2,