Week 1: Sections 6.1 and 6.2
Week 1
Ali Mousavidehshikh
Department of Mathematics University of Toronto
Ali Mousavidehshikh
Week 1
Outline
Week 1: Sections 6.1 and 6.2
1 Week 1: Sections 6.1 and 6.2
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
A vector space V is a non-empty set consisting of objects (called vectors) satisfying the following axioms for all u,v,w ∈ V, a,b ∈ R:
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
A vector space V is a non-empty set consisting of objects (called vectors) satisfying the following axioms for all u,v,w ∈ V, a,b ∈ R:
1. u + v ∈ V (closure),
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
A vector space V is a non-empty set consisting of objects (called vectors) satisfying the following axioms for all u,v,w ∈ V, a,b ∈ R:
1. u + v ∈ V (closure),
2. u + v = v + u (commutativity),
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
A vector space V is a non-empty set consisting of objects (called vectors) satisfying the following axioms for all u,v,w ∈ V, a,b ∈ R:
1. u + v ∈ V (closure),
2. u + v = v + u (commutativity),
3. u+(v+w)=(u+v)+w (associativity),
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
A vector space V is a non-empty set consisting of objects (called vectors) satisfying the following axioms for all u,v,w ∈ V, a,b ∈ R:
1. u + v ∈ V (closure),
2. u + v = v + u (commutativity),
3. u+(v+w)=(u+v)+w (associativity),
4. there exists 0 ∈ V such that u + 0 = u (zero element),
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
A vector space V is a non-empty set consisting of objects (called vectors) satisfying the following axioms for all u,v,w ∈ V, a,b ∈ R:
1. u + v ∈ V (closure),
2. u + v = v + u (commutativity),
3. u+(v+w)=(u+v)+w (associativity),
4. there exists 0 ∈ V such that u + 0 = u (zero element),
5. foreachv∈V,thereexistsau∈V suchthatv+u=0(uis
denoted by −v, additive inverse),
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
A vector space V is a non-empty set consisting of objects (called vectors) satisfying the following axioms for all u,v,w ∈ V, a,b ∈ R:
1. u + v ∈ V (closure),
2. u + v = v + u (commutativity),
3. u+(v+w)=(u+v)+w (associativity),
4. there exists 0 ∈ V such that u + 0 = u (zero element),
5. foreachv∈V,thereexistsau∈V suchthatv+u=0(uis
denoted by −v, additive inverse),
6. av ∈ V ,
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
A vector space V is a non-empty set consisting of objects (called vectors) satisfying the following axioms for all u,v,w ∈ V, a,b ∈ R:
1. u + v ∈ V (closure),
2. u + v = v + u (commutativity),
3. u+(v+w)=(u+v)+w (associativity),
4. there exists 0 ∈ V such that u + 0 = u (zero element),
5. foreachv∈V,thereexistsau∈V suchthatv+u=0(uis
denoted by −v, additive inverse),
6. av ∈ V ,
7. a(v+w)=av+aw,
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
A vector space V is a non-empty set consisting of objects (called vectors) satisfying the following axioms for all u,v,w ∈ V, a,b ∈ R:
1. u + v ∈ V (closure),
2. u + v = v + u (commutativity),
3. u+(v+w)=(u+v)+w (associativity),
4. there exists 0 ∈ V such that u + 0 = u (zero element),
5. foreachv∈V,thereexistsau∈V suchthatv+u=0(uis
denoted by −v, additive inverse),
6. av ∈ V ,
7. a(v+w)=av+aw,
8. (a+b)v=av+bv,
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
A vector space V is a non-empty set consisting of objects (called vectors) satisfying the following axioms for all u,v,w ∈ V, a,b ∈ R:
1. u + v ∈ V (closure),
2. u + v = v + u (commutativity),
3. u+(v+w)=(u+v)+w (associativity),
4. there exists 0 ∈ V such that u + 0 = u (zero element),
5. foreachv∈V,thereexistsau∈V suchthatv+u=0(uis
denoted by −v, additive inverse),
6. av ∈ V ,
7. a(v+w)=av+aw,
8. (a+b)v=av+bv,
9. a(bv) = (av)v,
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
A vector space V is a non-empty set consisting of objects (called vectors) satisfying the following axioms for all u,v,w ∈ V, a,b ∈ R:
1. u + v ∈ V (closure),
2. u + v = v + u (commutativity),
3. u+(v+w)=(u+v)+w (associativity),
4. there exists 0 ∈ V such that u + 0 = u (zero element),
5. foreachv∈V,thereexistsau∈V suchthatv+u=0(uis
denoted by −v, additive inverse),
6. av ∈ V ,
7. a(v+w)=av+aw,
8. (a+b)v=av+bv,
9. a(bv) = (av)v,
10. 1v = v.
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
Examples:
1. Rn under usual vector addition and scalar multiplication:
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
Examples:
1. Rn under usual vector addition and scalar multiplication:
(x1,…,xn)+(y1,…,yn) = (x1 +y1,…,xn +yn)
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
Examples:
1. Rn under usual vector addition and scalar multiplication:
(x1,…,xn)+(y1,…,yn) = (x1 +y1,…,xn +yn) a(x1,x2,…,xn) = (ax1,ax2,…,axn).
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
Examples:
1. Rn under usual vector addition and scalar multiplication:
(x1,…,xn)+(y1,…,yn) = (x1 +y1,…,xn +yn) a(x1,x2,…,xn) = (ax1,ax2,…,axn).
2. Mmn under usual matrix addition and scalar multiplication.
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
Examples:
1. Rn under usual vector addition and scalar multiplication:
(x1,…,xn)+(y1,…,yn) = (x1 +y1,…,xn +yn) a(x1,x2,…,xn) = (ax1,ax2,…,axn).
2. Mmn under usual matrix addition and scalar multiplication.
3. F [a, b] = {f : [a, b] → R a function} under pointwise addition
and scalar:
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
Examples:
1. Rn under usual vector addition and scalar multiplication:
(x1,…,xn)+(y1,…,yn) = (x1 +y1,…,xn +yn) a(x1,x2,…,xn) = (ax1,ax2,…,axn).
2. Mmn under usual matrix addition and scalar multiplication.
3. F [a, b] = {f : [a, b] → R a function} under pointwise addition
and scalar:multiplication:
(f +g)(x)=f(x)+f(y)and(af)(x)=a(f(x)).
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
Examples:
1. Rn under usual vector addition and scalar multiplication:
(x1,…,xn)+(y1,…,yn) = (x1 +y1,…,xn +yn) a(x1,x2,…,xn) = (ax1,ax2,…,axn).
2. Mmn under usual matrix addition and scalar multiplication.
3. F [a, b] = {f : [a, b] → R a function} under pointwise addition
and scalar:multiplication:
(f +g)(x)=f(x)+f(y)and(af)(x)=a(f(x)).
4. Fix n ∈ N∪{0}.
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
Examples:
1. Rn under usual vector addition and scalar multiplication:
(x1,…,xn)+(y1,…,yn) = (x1 +y1,…,xn +yn) a(x1,x2,…,xn) = (ax1,ax2,…,axn).
2. Mmn under usual matrix addition and scalar multiplication.
3. F [a, b] = {f : [a, b] → R a function} under pointwise addition
and scalar:multiplication:
(f +g)(x)=f(x)+f(y)and(af)(x)=a(f(x)).
4. Fixn∈N∪{0}.Then,Pn ={a0+a1x+…anxn :ai ∈R}isa vector space under addition and scalar multiplication defined on page 291 in your book.
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
Examples:
1. Rn under usual vector addition and scalar multiplication:
(x1,…,xn)+(y1,…,yn) = (x1 +y1,…,xn +yn) a(x1,x2,…,xn) = (ax1,ax2,…,axn).
2. Mmn under usual matrix addition and scalar multiplication.
3. F [a, b] = {f : [a, b] → R a function} under pointwise addition
and scalar:multiplication:
(f +g)(x)=f(x)+f(y)and(af)(x)=a(f(x)).
4. Fixn∈N∪{0}.Then,Pn ={a0+a1x+…anxn :ai ∈R}isa vector space under addition and scalar multiplication defined on page 291 in your book.
5. The set of all polynomials, denoted by P, is a vector space (under the operations defined on page 291 in your book).
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
Examples:
1. Rn under usual vector addition and scalar multiplication:
(x1,…,xn)+(y1,…,yn) = (x1 +y1,…,xn +yn) a(x1,x2,…,xn) = (ax1,ax2,…,axn).
2. Mmn under usual matrix addition and scalar multiplication.
3. F [a, b] = {f : [a, b] → R a function} under pointwise addition
and scalar:multiplication:
(f +g)(x)=f(x)+f(y)and(af)(x)=a(f(x)).
4. Fixn∈N∪{0}.Then,Pn ={a0+a1x+…anxn :ai ∈R}isa vector space under addition and scalar multiplication defined on page 291 in your book.
5. The set of all polynomials, denoted by P, is a vector space (under the operations defined on page 291 in your book).
Question. Is R2 a vector space under the operations
(a, b) + (c, d) = (a + c − 1, b + d + 1) and z(a, b) = (zb, za)?
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
We define the difference of two vectors v and u in a vector spaceV asfollows: v−u=v+(−u).
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
We define the difference of two vectors v and u in a vector spaceV asfollows: v−u=v+(−u).
(Theorem) Let v denote a vector in a vector space and let a ∈ R and 0 be the zero element in V .
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
We define the difference of two vectors v and u in a vector spaceV asfollows: v−u=v+(−u).
(Theorem) Let v denote a vector in a vector space and let a ∈ R and 0 be the zero element in V.Then
1. 0v = 0, where 0 is the real number 0.
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
We define the difference of two vectors v and u in a vector spaceV asfollows: v−u=v+(−u).
(Theorem) Let v denote a vector in a vector space and let a ∈ R and 0 be the zero element in V.Then
1. 0v = 0, where 0 is the real number 0. 2. a0=0.
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
We define the difference of two vectors v and u in a vector spaceV asfollows: v−u=v+(−u).
(Theorem) Let v denote a vector in a vector space and let a ∈ R and 0 be the zero element in V.Then
1. 0v = 0, where 0 is the real number 0. 2. a0=0.
3. Ifav=0,thena=0orv=0.
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
We define the difference of two vectors v and u in a vector spaceV asfollows: v−u=v+(−u).
(Theorem) Let v denote a vector in a vector space and let a ∈ R and 0 be the zero element in V.Then
1. 0v = 0, where 0 is the real number 0. 2. a0=0.
3. Ifav=0,thena=0orv=0.
4. (−1)v = −v.
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
We define the difference of two vectors v and u in a vector spaceV asfollows: v−u=v+(−u).
(Theorem) Let v denote a vector in a vector space and let a ∈ R and 0 be the zero element in V.Then
1. 0v = 0, where 0 is the real number 0. 2. a0=0.
3. Ifav=0,thena=0orv=0.
4. (−1)v = −v.
5. (−a)v = −(−av) = a(−v).
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
We define the difference of two vectors v and u in a vector spaceV asfollows: v−u=v+(−u).
(Theorem) Let v denote a vector in a vector space and let a ∈ R and 0 be the zero element in V.Then
1. 0v = 0, where 0 is the real number 0. 2. a0=0.
3. Ifav=0,thena=0orv=0.
4. (−1)v = −v.
5. (−a)v = −(−av) = a(−v). 6. 0 is unique.
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
We define the difference of two vectors v and u in a vector spaceV asfollows: v−u=v+(−u).
(Theorem) Let v denote a vector in a vector space and let a ∈ R and 0 be the zero element in V.Then
1. 0v = 0, where 0 is the real number 0. 2. a0=0.
3. Ifav=0,thena=0orv=0.
4. (−1)v = −v.
5. (−a)v = −(−av) = a(−v). 6. 0 is unique.
7. −v is unique.
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
(Definition) If V is a vector space, a non-empty subset U ⊆ V is called a subspace of V if U is itself a vector space using the addition and scalar multiplication of V .
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
(Definition) If V is a vector space, a non-empty subset U ⊆ V is called a subspace of V if U is itself a vector space using the addition and scalar multiplication of V .
(Theorem) A subset U of a vector space V is a subspace of V if it satisfies the following three conditions:
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
(Definition) If V is a vector space, a non-empty subset U ⊆ V is called a subspace of V if U is itself a vector space using the addition and scalar multiplication of V .
(Theorem) A subset U of a vector space V is a subspace of V if it satisfies the following three conditions:
1. 0liesinU,where0isthezerovectorinV.
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
(Definition) If V is a vector space, a non-empty subset U ⊆ V is called a subspace of V if U is itself a vector space using the addition and scalar multiplication of V .
(Theorem) A subset U of a vector space V is a subspace of V if it satisfies the following three conditions:
1. 0liesinU,where0isthezerovectorinV. 2. Ifv,u∈U,thenv+u∈U.
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
(Definition) If V is a vector space, a non-empty subset U ⊆ V is called a subspace of V if U is itself a vector space using the addition and scalar multiplication of V .
(Theorem) A subset U of a vector space V is a subspace of V if it satisfies the following three conditions:
1. 0liesinU,where0isthezerovectorinV. 2. Ifv,u∈U,thenv+u∈U.
3. Ifu∈U anda∈R,thenau∈U.
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
(Definition) If V is a vector space, a non-empty subset U ⊆ V is called a subspace of V if U is itself a vector space using the addition and scalar multiplication of V .
(Theorem) A subset U of a vector space V is a subspace of V if it satisfies the following three conditions:
1. 0liesinU,where0isthezerovectorinV. 2. Ifv,u∈U,thenv+u∈U.
3. Ifu∈U anda∈R,thenau∈U.
(Example) Let v be a vector in a vector space V . The set Rv = {av : a ∈ R} is subspace of V .
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
(Definition) If V is a vector space, a non-empty subset U ⊆ V is called a subspace of V if U is itself a vector space using the addition and scalar multiplication of V .
(Theorem) A subset U of a vector space V is a subspace of V if it satisfies the following three conditions:
1. 0liesinU,where0isthezerovectorinV. 2. Ifv,u∈U,thenv+u∈U.
3. Ifu∈U anda∈R,thenau∈U.
(Example) Let v be a vector in a vector space V . The set Rv = {av : a ∈ R} is subspace of V .
(Example) Pn is a subspace of P for each n ∈ N ∪ {0}.
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
(Definition) Let {v1, v2, . . . , vn} be a set of vectors in a vector space V . Then
span{v1,v2,…,vn}={a1v1 +a2v2 +…+anvn :ai ∈R}.
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
(Definition) Let {v1, v2, . . . , vn} be a set of vectors in a vector space V . Then
span{v1,v2,…,vn}={a1v1 +a2v2 +…+anvn :ai ∈R}.
If V = span{v1,v2,…,vn}, then we say that {v1,v2,…,vn} is a spanning set for V .
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
(Definition) Let {v1, v2, . . . , vn} be a set of vectors in a vector space V . Then
span{v1,v2,…,vn}={a1v1 +a2v2 +…+anvn :ai ∈R}.
If V = span{v1,v2,…,vn}, then we say that {v1,v2,…,vn}
is a spanning set for V .
(Example) Pn = span{1,x,x2,…,xn}.
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
(Definition) Let {v1, v2, . . . , vn} be a set of vectors in a vector space V . Then
span{v1,v2,…,vn}={a1v1 +a2v2 +…+anvn :ai ∈R}.
If V = span{v1,v2,…,vn}, then we say that {v1,v2,…,vn}
is a spanning set for V .
(Example) Pn = span{1,x,x2,…,xn}. (Example) P = span{1,x,x2,x3,x4,…}.
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
(Definition) Let {v1, v2, . . . , vn} be a set of vectors in a vector space V . Then
span{v1,v2,…,vn}={a1v1 +a2v2 +…+anvn :ai ∈R}.
If V = span{v1,v2,…,vn}, then we say that {v1,v2,…,vn}
is a spanning set for V .
(Example) Pn = span{1,x,x2,…,xn}. (Example) P = span{1,x,x2,x3,x4,…}. (Example) R3 = span{(1, 0, 0), (0, 1, 0), (0, 0, 1)}.
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
(Definition) Let {v1, v2, . . . , vn} be a set of vectors in a vector space V . Then
span{v1,v2,…,vn}={a1v1 +a2v2 +…+anvn :ai ∈R}.
If V = span{v1,v2,…,vn}, then we say that {v1,v2,…,vn}
is a spanning set for V .
(Example) Pn = span{1,x,x2,…,xn}.
(Example) P = span{1,x,x2,x3,x4,…}.
(Example) R3 = span{(1, 0, 0), (0, 1, 0), (0, 0, 1)}.Notice that R3 = span{(1, 0, 0), (1, 1, 0), (1, 1, 1)}.
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
(Definition) Let {v1, v2, . . . , vn} be a set of vectors in a vector space V . Then
span{v1,v2,…,vn}={a1v1 +a2v2 +…+anvn :ai ∈R}.
If V = span{v1,v2,…,vn}, then we say that {v1,v2,…,vn}
is a spanning set for V .
(Example) Pn = span{1,x,x2,…,xn}.
(Example) P = span{1,x,x2,x3,x4,…}.
(Example) R3 = span{(1, 0, 0), (0, 1, 0), (0, 0, 1)}.Notice that R3 = span{(1, 0, 0), (1, 1, 0), (1, 1, 1)}.
A vector space can have many sets that span it.
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
(Example) Let u and v be in a vector space. Then span{u, v } = span{u + v , u − v }.
Week 1
Ali Mousavidehshikh
Week 1: Sections 6.1 and 6.2
(Example) Let u and v be in a vector space. Then span{u, v } = span{u + v , u − v }.
(Theorem) Let U = span{v1, v2, . . . , vn} in a vector space V.
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
(Example) Let u and v be in a vector space. Then span{u, v } = span{u + v , u − v }.
(Theorem) Let U = span{v1, v2, . . . , vn} in a vector space V. Then
1. U is a subspace of V.
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
(Example) Let u and v be in a vector space. Then span{u, v } = span{u + v , u − v }.
(Theorem) Let U = span{v1, v2, . . . , vn} in a vector space V. Then
1. U is a subspace of V.
2. U contains each of v1,v2,…,vn.
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
(Example) Let u and v be in a vector space. Then span{u, v } = span{u + v , u − v }.
(Theorem) Let U = span{v1, v2, . . . , vn} in a vector space V. Then
1. U is a subspace of V.
2. U contains each of v1,v2,…,vn.
3. U is the smallest subspace containing the vectors
v1,v2,…,vn;
Ali Mousavidehshikh
Week 1
Week 1: Sections 6.1 and 6.2
(Example) Let u and v be in a vector space. Then span{u, v } = span{u + v , u − v }.
(Theorem) Let U = span{v1, v2, . . . , vn} in a vector space V. Then
1. U is a subspace of V.
2. U contains each of v1,v2,…,vn.
3. U is the smallest subspace containing the vectors
v1, v2, . . . , vn;in the sense that any subspace of V containing v1,v2,…,vn must contain all of U.
Ali Mousavidehshikh
Week 1