Vector Spaces
Shahid Hussain
March 16, 2009
Shahid Hussain () Vector Spaces March 16, 2009 1 / 17
Outline
1 Spaces and SubspacesSpacesSubspaces
Shahid Hussain () Vector Spaces March 16, 2009 2 / 17
Spaces and Subspaces Spaces
Introduction
Matrix theory became established toward the end of the nineteenthcentury,
Different mathematical entities that were considered to be quitedifferent from matrices were in fact quite similar. For example:
Points in the plane R2,points in 3-space R3,polynomials,continuous functions, anddifferentiable functions (to name only a few)
were recognized to satisfy the same additive properties and scalarmultiplication properties for matrices.
Rather than studying each topic separately, it was reasoned that it ismore efficient and productive to study many topics at one time bystudying the common properties that they satisfy.
This eventually led to the axiomatic definition of a vector space.
Shahid Hussain () Vector Spaces March 16, 2009 3 / 17
Spaces and Subspaces Spaces
Introduction
Matrix theory became established toward the end of the nineteenthcentury,
Different mathematical entities that were considered to be quitedifferent from matrices were in fact quite similar. For example:
Points in the plane R2,points in 3-space R3,polynomials,continuous functions, anddifferentiable functions (to name only a few)
were recognized to satisfy the same additive properties and scalarmultiplication properties for matrices.
Rather than studying each topic separately, it was reasoned that it ismore efficient and productive to study many topics at one time bystudying the common properties that they satisfy.
This eventually led to the axiomatic definition of a vector space.
Shahid Hussain () Vector Spaces March 16, 2009 3 / 17
Spaces and Subspaces Spaces
Introduction
Matrix theory became established toward the end of the nineteenthcentury,
Different mathematical entities that were considered to be quitedifferent from matrices were in fact quite similar. For example:
Points in the plane R2,
points in 3-space R3,polynomials,continuous functions, anddifferentiable functions (to name only a few)
were recognized to satisfy the same additive properties and scalarmultiplication properties for matrices.
Rather than studying each topic separately, it was reasoned that it ismore efficient and productive to study many topics at one time bystudying the common properties that they satisfy.
This eventually led to the axiomatic definition of a vector space.
Shahid Hussain () Vector Spaces March 16, 2009 3 / 17
Spaces and Subspaces Spaces
Introduction
Matrix theory became established toward the end of the nineteenthcentury,
Different mathematical entities that were considered to be quitedifferent from matrices were in fact quite similar. For example:
Points in the plane R2,points in 3-space R3,
polynomials,continuous functions, anddifferentiable functions (to name only a few)
were recognized to satisfy the same additive properties and scalarmultiplication properties for matrices.
Rather than studying each topic separately, it was reasoned that it ismore efficient and productive to study many topics at one time bystudying the common properties that they satisfy.
This eventually led to the axiomatic definition of a vector space.
Shahid Hussain () Vector Spaces March 16, 2009 3 / 17
Spaces and Subspaces Spaces
Introduction
Matrix theory became established toward the end of the nineteenthcentury,
Different mathematical entities that were considered to be quitedifferent from matrices were in fact quite similar. For example:
Points in the plane R2,points in 3-space R3,polynomials,
continuous functions, anddifferentiable functions (to name only a few)
were recognized to satisfy the same additive properties and scalarmultiplication properties for matrices.
Rather than studying each topic separately, it was reasoned that it ismore efficient and productive to study many topics at one time bystudying the common properties that they satisfy.
This eventually led to the axiomatic definition of a vector space.
Shahid Hussain () Vector Spaces March 16, 2009 3 / 17
Spaces and Subspaces Spaces
Introduction
Matrix theory became established toward the end of the nineteenthcentury,
Different mathematical entities that were considered to be quitedifferent from matrices were in fact quite similar. For example:
Points in the plane R2,points in 3-space R3,polynomials,continuous functions, and
differentiable functions (to name only a few)
were recognized to satisfy the same additive properties and scalarmultiplication properties for matrices.
Rather than studying each topic separately, it was reasoned that it ismore efficient and productive to study many topics at one time bystudying the common properties that they satisfy.
This eventually led to the axiomatic definition of a vector space.
Shahid Hussain () Vector Spaces March 16, 2009 3 / 17
Spaces and Subspaces Spaces
Introduction
Matrix theory became established toward the end of the nineteenthcentury,
Different mathematical entities that were considered to be quitedifferent from matrices were in fact quite similar. For example:
Points in the plane R2,points in 3-space R3,polynomials,continuous functions, anddifferentiable functions (to name only a few)
were recognized to satisfy the same additive properties and scalarmultiplication properties for matrices.
Rather than studying each topic separately, it was reasoned that it ismore efficient and productive to study many topics at one time bystudying the common properties that they satisfy.
This eventually led to the axiomatic definition of a vector space.
Shahid Hussain () Vector Spaces March 16, 2009 3 / 17
Spaces and Subspaces Spaces
Introduction
Matrix theory became established toward the end of the nineteenthcentury,
Different mathematical entities that were considered to be quitedifferent from matrices were in fact quite similar. For example:
Points in the plane R2,points in 3-space R3,polynomials,continuous functions, anddifferentiable functions (to name only a few)
were recognized to satisfy the same additive properties and scalarmultiplication properties for matrices.
Rather than studying each topic separately, it was reasoned that it ismore efficient and productive to study many topics at one time bystudying the common properties that they satisfy.
This eventually led to the axiomatic definition of a vector space.
Shahid Hussain () Vector Spaces March 16, 2009 3 / 17
Spaces and Subspaces Spaces
Introduction
Matrix theory became established toward the end of the nineteenthcentury,
Different mathematical entities that were considered to be quitedifferent from matrices were in fact quite similar. For example:
Points in the plane R2,points in 3-space R3,polynomials,continuous functions, anddifferentiable functions (to name only a few)
were recognized to satisfy the same additive properties and scalarmultiplication properties for matrices.
Rather than studying each topic separately, it was reasoned that it ismore efficient and productive to study many topics at one time bystudying the common properties that they satisfy.
This eventually led to the axiomatic definition of a vector space.
Shahid Hussain () Vector Spaces March 16, 2009 3 / 17
Spaces and Subspaces Spaces
Vector Space
Vector Space
A vector space involves four things-two sets V and F , and two algebraicoperations called vector addition and scalar multiplication.
V is a nonempty set of objects called vectors. Although V can bequite general, we will usually consider V to be a set of n-tuples or aset of matrices.
F is a scalar field-for us F is either the field R of real numbers or thefield C of complex numbers.Vector addition (denoted by x+ y) is an operation between elementsof V.Scalar multiplication (denoted by x) is an operation betweenelements of F and V.
Shahid Hussain () Vector Spaces March 16, 2009 4 / 17
Spaces and Subspaces Spaces
Vector Space
Vector Space
A vector space involves four things-two sets V and F , and two algebraicoperations called vector addition and scalar multiplication.
V is a nonempty set of objects called vectors. Although V can bequite general, we will usually consider V to be a set of n-tuples or aset of matrices.
F is a scalar field-for us F is either the field R of real numbers or thefield C of complex numbers.
Vector addition (denoted by x+ y) is an operation between elementsof V.Scalar multiplication (denoted by x) is an operation betweenelements of F and V.
Shahid Hussain () Vector Spaces March 16, 2009 4 / 17
Spaces and Subspaces Spaces
Vector Space
Vector Space
A vector space involves four things-two sets V and F , and two algebraicoperations called vector addition and scalar multiplication.
V is a nonempty set of objects called vectors. Although V can bequite general, we will usually consider V to be a set of n-tuples or aset of matrices.
F is a scalar field-for us F is either the field R of real numbers or thefield C of complex numbers.Vector addition (denoted by x+ y) is an operation between elementsof V.
Scalar multiplication (denoted by x) is an operation betweenelements of F and V.
Shahid Hussain () Vector Spaces March 16, 2009 4 / 17
Spaces and Subspaces Spaces
Vector Space
Vector Space
A vector space involves four things-two sets V and F , and two algebraicoperations called vector addition and scalar multiplication.
V is a nonempty set of objects called vectors. Although V can bequite general, we will usually consider V to be a set of n-tuples or aset of matrices.
F is a scalar field-for us F is either the field R of real numbers or thefield C of complex numbers.Vector addition (denoted by x+ y) is an operation between elementsof V.Scalar multiplication (denoted by x) is an operation betweenelements of F and V.
Shahid Hussain () Vector Spaces March 16, 2009 4 / 17
Spaces and Subspaces Spaces
Cont.
Field
Let F be a subset of the real numbers R (or the complex numbers C). Weshall say that F is a field if it satisfies the following conditions:
1 If x, y are elements of F , then x+ y and xy are also elements of F .
2 If x F , then x is also an element of F . If furthermore x 6= 0 thenx1 is an element of F .
3 The elements 0 and 1 are elements of F .
For example that both R and C are fields. Verify!The set of rational numbers Q is also a field. Verify!The set of all integers Z is not a field. Why?
Shahid Hussain () Vector Spaces March 16, 2009 5 / 17
Spaces and Subspaces Spaces
Cont.
Field
Let F be a subset of the real numbers R (or the complex numbers C). Weshall say that F is a field if it satisfies the following conditions:
1 If x, y are elements of F , then x+ y and xy are also elements of F .2 If x F , then x is also an element of F . If furthermore x 6= 0 thenx1 is an element of F .
3 The elements 0 and 1 are elements of F .
For example that both R and C are fields. Verify!The set of rational numbers Q is also a field. Verify!The set of all integers Z is not a field. Why?
Shahid Hussain () Vector Spaces March 16, 2009 5 / 17
Spaces and Subspaces Spaces
Cont.
Field
Let F be a subset of the real numbers R (or the complex numbers C). Weshall say that F is a field if it satisfies the following conditions:
1 If x, y are elements of F , then x+ y and xy are also elements of F .2 If x F , then x is also an element of F . If furthermore x 6= 0 thenx1 is an element of F .
3 The elements 0 and 1 are elements of F .
For example that both R and C are fields. Verify!The set of rational numbers Q is also a field. Verify!The set of all integers Z is not a field. Why?
Shahid Hussain () Vector Spaces March 16, 2009 5 / 17
Spaces and Subspaces Spaces
Cont.
Field
Let F be a subset of the real numbers R (or the complex numbers C). Weshall say that F is a field if it satisfies the following conditions:
1 If x, y are elements of F , then x+ y and xy are also elements of F .2 If x F , then x is also an element of F . If furthermore x 6= 0 thenx1 is an element of F .
3 The elements 0 and 1 are elements of F .
For example that both R and C are fields. Verify!
The set of rational numbers Q is also a field. Verify!The set of all integers Z is not a field. Why?
Shahid Hussain () Vector Spaces March 16, 2009 5 / 17
Spaces and Subspaces Spaces
Cont.
Field
Let F be a subset of the real numbers R (or the complex numbers C). Weshall say that F is a field if it satisfies the following conditions:
1 If x, y are elements of F , then x+ y and xy are also elements of F .2 If x F , then x is also an element of F . If furthermore x 6= 0 thenx1 is an element of F .
3 The elements 0 and 1 are elements of F .
For example that both R and C are fields. Verify!The set of rational numbers Q is also a field. Verify!
The set of all integers Z is not a field. Why?
Shahid Hussain () Vector Spaces March 16, 2009 5 / 17
Spaces and Subspaces Spaces
Cont.
Field
Let F be a subset of the real numbers R (or the complex numbers C). Weshall say that F is a field if it satisfies the following conditions:
1 If x, y are elements of F , then x+ y and xy are also elements of F .2 If x F , then x is also an element of F . If furthermore x 6= 0 thenx1 is an element of F .
3 The elements 0 and 1 are elements of F .
For example that both R and C are fields. Verify!The set of rational numbers Q is also a field. Verify!The set of all integers Z is not a field. Why?
Shahid Hussain () Vector Spaces March 16, 2009 5 / 17
Spaces and Subspaces Spaces
Cont.
Vector Space
The set V is called a vector space over F when the vector addition andscalar multiplication operations satisfy the following properties.
Vector Addition
A1. x+ y V for all x,y V.
A2. (x+ y) + z = x+ (y + z) for everyx,y, z V
A3. x+ y = y + x for every x,y V.A4. There exits a 0 V such that
x+ 0 = x for every x V.A5. For each x V, there exits a
(x) V such that x+ (x) = 0.
Scalar Multiplication
M1. x V for all F and x V.M2. ()x = (x) for all , F and
every x V.M3. (x+ y) = x+ y for every F
and all x,y V.M4. (+ )x = x+ x for all , F
and every x V.M5. 1x = x for every x V.
Shahid Hussain () Vector Spaces March 16, 2009 6 / 17
Spaces and Subspaces Spaces
Cont.
Vector Space
The set V is called a vector space over F when the vector addition andscalar multiplication operations satisfy the following properties.
Vector Addition
A1. x+ y V for all x,y V.A2. (x+ y) + z = x+ (y + z) for every
x,y, z V
A3. x+ y = y + x for every x,y V.A4. There exits a 0 V such that
x+ 0 = x for every x V.A5. For each x V, there exits a
(x) V such that x+ (x) = 0.
Scalar Multiplication
M1. x V for all F and x V.M2. ()x = (x) for all , F and
every x V.M3. (x+ y) = x+ y for every F
and all x,y V.M4. (+ )x = x+ x for all , F
and every x V.M5. 1x = x for every x V.
Shahid Hussain () Vector Spaces March 16, 2009 6 / 17
Spaces and Subspaces Spaces
Cont.
Vector Space
The set V is called a vector space over F when the vector addition andscalar multiplication operations satisfy the following properties.
Vector Addition
A1. x+ y V for all x,y V.A2. (x+ y) + z = x+ (y + z) for every
x,y, z VA3. x+ y = y + x for every x,y V.
A4. There exits a 0 V such thatx+ 0 = x for every x V.
A5. For each x V, there exits a(x) V such that x+ (x) = 0.
Scalar Multiplication
M1. x V for all F and x V.M2. ()x = (x) for all , F and
every x V.M3. (x+ y) = x+ y for every F
and all x,y V.M4. (+ )x = x+ x for all , F
and every x V.M5. 1x = x for every x V.
Shahid Hussain () Vector Spaces March 16, 2009 6 / 17
Spaces and Subspaces Spaces
Cont.
Vector Space
The set V is called a vector space over F when the vector addition andscalar multiplication operations satisfy the following properties.
Vector Addition
A1. x+ y V for all x,y V.A2. (x+ y) + z = x+ (y + z) for every
x,y, z VA3. x+ y = y + x for every x,y V.A4. There exits a 0 V such that
x+ 0 = x for every x V.
A5. For each x V, there exits a(x) V such that x+ (x) = 0.
Scalar Multiplication
M1. x V for all F and x V.M2. ()x = (x) for all , F and
every x V.M3. (x+ y) = x+ y for every F
and all x,y V.M4. (+ )x = x+ x for all , F
and every x V.M5. 1x = x for every x V.
Shahid Hussain () Vector Spaces March 16, 2009 6 / 17
Spaces and Subspaces Spaces
Cont.
Vector Space
The set V is called a vector space over F when the vector addition andscalar multiplication operations satisfy the following properties.
Vector Addition
A1. x+ y V for all x,y V.A2. (x+ y) + z = x+ (y + z) for every
x,y, z VA3. x+ y = y + x for every x,y V.A4. There exits a 0 V such that
x+ 0 = x for every x V.A5. For each x V, there exits a
(x) V such that x+ (x) = 0.
Scalar Multiplication
M1. x V for all F and x V.M2. ()x = (x) for all , F and
every x V.M3. (x+ y) = x+ y for every F
and all x,y V.M4. (+ )x = x+ x for all , F
and every x V.M5. 1x = x for every x V.
Shahid Hussain () Vector Spaces March 16, 2009 6 / 17
Spaces and Subspaces Spaces
Cont.
Vector Space
The set V is called a vector space over F when the vector addition andscalar multiplication operations satisfy the following properties.
Vector Addition
A1. x+ y V for all x,y V.A2. (x+ y) + z = x+ (y + z) for every
x,y, z VA3. x+ y = y + x for every x,y V.A4. There exits a 0 V such that
x+ 0 = x for every x V.A5. For each x V, there exits a
(x) V such that x+ (x) = 0.
Scalar Multiplication
M1. x V for all F and x V.
M2. ()x = (x) for all , F andevery x V.
M3. (x+ y) = x+ y for every Fand all x,y V.
M4. (+ )x = x+ x for all , Fand every x V.
M5. 1x = x for every x V.
Shahid Hussain () Vector Spaces March 16, 2009 6 / 17
Spaces and Subspaces Spaces
Cont.
Vector Space
The set V is called a vector space over F when the vector addition andscalar multiplication operations satisfy the following properties.
Vector Addition
A1. x+ y V for all x,y V.A2. (x+ y) + z = x+ (y + z) for every
x,y, z VA3. x+ y = y + x for every x,y V.A4. There exits a 0 V such that
x+ 0 = x for every x V.A5. For each x V, there exits a
(x) V such that x+ (x) = 0.
Scalar Multiplication
M1. x V for all F and x V.M2. ()x = (x) for all , F and
every x V.
M3. (x+ y) = x+ y for every Fand all x,y V.
M4. (+ )x = x+ x for all , Fand every x V.
M5. 1x = x for every x V.
Shahid Hussain () Vector Spaces March 16, 2009 6 / 17
Spaces and Subspaces Spaces
Cont.
Vector Space
The set V is called a vector space over F when the vector addition andscalar multiplication operations satisfy the following properties.
Vector Addition
A1. x+ y V for all x,y V.A2. (x+ y) + z = x+ (y + z) for every
x,y, z VA3. x+ y = y + x for every x,y V.A4. There exits a 0 V such that
x+ 0 = x for every x V.A5. For each x V, there exits a
(x) V such that x+ (x) = 0.
Scalar Multiplication
M1. x V for all F and x V.M2. ()x = (x) for all , F and
every x V.M3. (x+ y) = x+ y for every F
and all x,y V.
M4. (+ )x = x+ x for all , Fand every x V.
M5. 1x = x for every x V.
Shahid Hussain () Vector Spaces March 16, 2009 6 / 17
Spaces and Subspaces Spaces
Cont.
Vector Space
The set V is called a vector space over F when the vector addition andscalar multiplication operations satisfy the following properties.
Vector Addition
A1. x+ y V for all x,y V.A2. (x+ y) + z = x+ (y + z) for every
x,y, z VA3. x+ y = y + x for every x,y V.A4. There exits a 0 V such that
x+ 0 = x for every x V.A5. For each x V, there exits a
(x) V such that x+ (x) = 0.
Scalar Multiplication
M1. x V for all F and x V.M2. ()x = (x) for all , F and
every x V.M3. (x+ y) = x+ y for every F
and all x,y V.M4. (+ )x = x+ x for all , F
and every x V.
M5. 1x = x for every x V.
Shahid Hussain () Vector Spaces March 16, 2009 6 / 17
Spaces and Subspaces Spaces
Cont.
Vector Space
The set V is called a vector space over F when the vector addition andscalar multiplication operations satisfy the following properties.
Vector Addition
A1. x+ y V for all x,y V.A2. (x+ y) + z = x+ (y + z) for every
x,y, z VA3. x+ y = y + x for every x,y V.A4. There exits a 0 V such that
x+ 0 = x for every x V.A5. For each x V, there exits a
(x) V such that x+ (x) = 0.
Scalar Multiplication
M1. x V for all F and x V.M2. ()x = (x) for all , F and
every x V.M3. (x+ y) = x+ y for every F
and all x,y V.M4. (+ )x = x+ x for all , F
and every x V.M5. 1x = x for every x V.
Shahid Hussain () Vector Spaces March 16, 2009 6 / 17
Spaces and Subspaces Spaces
Example 1
Because A1.A5. are generalized versions of the five additive properties ofmatrix addition, and M1.M5. are generalizations of the five scalarmultiplication properties, we can say that the following hold.
The set Rmn of m n real matrices is a vector space over R.
The set Cmn of m n complex matrices is a vector space over C.
Shahid Hussain () Vector Spaces March 16, 2009 7 / 17
Spaces and Subspaces Spaces
Example 1
Because A1.A5. are generalized versions of the five additive properties ofmatrix addition, and M1.M5. are generalizations of the five scalarmultiplication properties, we can say that the following hold.
The set Rmn of m n real matrices is a vector space over R.The set Cmn of m n complex matrices is a vector space over C.
Shahid Hussain () Vector Spaces March 16, 2009 7 / 17
Spaces and Subspaces Spaces
Example 2: The real coordinate spaces
The real coordinate spaces
R1n = {(x1 x2 xn), xi R} and Rn1 =
x1x2...xn
, xi R
are special cases of the preceding example, and these will be the object ofmost of our attention. In the context of vector spaces, it usually makes nodifference whether a coordinate vector is depicted as a row or as a column.Therefore we will use the common symbol Rn to designate a coordinatespace. When it is important to distinguish between rows and columns, wewill explicitly write R1n of Rn1. Similar remarks hold for complexcoordinate spaces.
Shahid Hussain () Vector Spaces March 16, 2009 8 / 17
Spaces and Subspaces Spaces
Example 3
With function addition and scalar multiplication defined by
(f + g)(x) = f(x) + g(x) and (f)(x) = f(x),
the following sets are vector spaces over R:The set of functions mapping the interval [0, 1] into R.
The set of all real-valued continuous functions defined on [0, 1].The set of real-valued functions that are differentiable on [0, 1].The set of all polynomials with real coefficients.
Shahid Hussain () Vector Spaces March 16, 2009 9 / 17
Spaces and Subspaces Spaces
Example 3
With function addition and scalar multiplication defined by
(f + g)(x) = f(x) + g(x) and (f)(x) = f(x),
the following sets are vector spaces over R:The set of functions mapping the interval [0, 1] into R.The set of all real-valued continuous functions defined on [0, 1].
The set of real-valued functions that are differentiable on [0, 1].The set of all polynomials with real coefficients.
Shahid Hussain () Vector Spaces March 16, 2009 9 / 17
Spaces and Subspaces Spaces
Example 3
With function addition and scalar multiplication defined by
(f + g)(x) = f(x) + g(x) and (f)(x) = f(x),
the following sets are vector spaces over R:The set of functions mapping the interval [0, 1] into R.The set of all real-valued continuous functions defined on [0, 1].The set of real-valued functions that are differentiable on [0, 1].
The set of all polynomials with real coefficients.
Shahid Hussain () Vector Spaces March 16, 2009 9 / 17
Spaces and Subspaces Spaces
Example 3
With function addition and scalar multiplication defined by
(f + g)(x) = f(x) + g(x) and (f)(x) = f(x),
the following sets are vector spaces over R:The set of functions mapping the interval [0, 1] into R.The set of all real-valued continuous functions defined on [0, 1].The set of real-valued functions that are differentiable on [0, 1].The set of all polynomials with real coefficients.
Shahid Hussain () Vector Spaces March 16, 2009 9 / 17
Spaces and Subspaces Spaces
Example 4
Consider the vector space R2, and let
L = {(x, y)|y = x}
be a line through the origin. L is a subset of R2, but L is a special kind ofsubset because L also satisfies the properties A1.A5. and M1.M5.that define a vector space. This shows that it is possible for one vectorspace to properly contain other vector spaces.
Shahid Hussain () Vector Spaces March 16, 2009 10 / 17
Spaces and Subspaces Subspaces
Subspace
Subspace
Let S be a nonempty subset of a vector space V over F (symbolically,S V). If S is also a vector space over F using the same addition That is,a nonempty subset S of a vector space V is a subspace of V if and only ifA1. x,y S x+ y S, and
M1. x S x S for all F .
Shahid Hussain () Vector Spaces March 16, 2009 11 / 17
Spaces and Subspaces Subspaces
Subspace
Subspace
Let S be a nonempty subset of a vector space V over F (symbolically,S V). If S is also a vector space over F using the same addition That is,a nonempty subset S of a vector space V is a subspace of V if and only ifA1. x,y S x+ y S, andM1. x S x S for all F .
Shahid Hussain () Vector Spaces March 16, 2009 11 / 17
Spaces and Subspaces Subspaces
Example 1
Given a vector space V, the set Z = {0} containing only the zero vector isa subspace of V because A1. and M1. are trivially satisfied. Naturally, thissubspace is called the trivial subspace.
Shahid Hussain () Vector Spaces March 16, 2009 12 / 17
Spaces and Subspaces Subspaces
Linear Combination
Linear Combination
Let V be an arbitrary vector space, and let v1,v2, . . . ,vr be elements ofV. Let 1, 2, . . . , r be scalars. An expression of type
1v1 + 2v2 + . . .+ rvr
is called a linear combination of v1,v2, . . . ,vr.
Shahid Hussain () Vector Spaces March 16, 2009 13 / 17
Spaces and Subspaces Subspaces
Spanning
Spanning
For a set of vectors S = {v1,v2, . . . ,vr} from a vector space V, the set ofall possible linear combinations of the vis is denoted by
span(S) = {1v1 + 2v2 + + rvr|i F}.
Notice that span(S) is a subspace of V because the two closure propertiesA1. and M1. are satisfied. That is, if x =
i ivi and y =
i ivi are
two linear combinations from span(S), then the sumx+ y =
i(i + i)vi is also a linear combination in span(S), and for any
scalar , x =
i(i)vi is also a linear combination in span(S).
Shahid Hussain () Vector Spaces March 16, 2009 14 / 17
Spaces and Subspaces Subspaces
Spanning Sets
Spanning Sets
For a set of vectors S = {v1,v2, . . . ,vr} , the subspacespan(S) = {1v1 + 2v2 + + rvr} generated by forming alllinear combinations of vectors from S is called the space spanned byS.
If V is a vector space such that V = span(S), we say S is a spanningset for V. In other words, S spans V whenever each vector in V is alinear combination of vectors from S.
Shahid Hussain () Vector Spaces March 16, 2009 15 / 17
Spaces and Subspaces Subspaces
Spanning Sets
Spanning Sets
For a set of vectors S = {v1,v2, . . . ,vr} , the subspacespan(S) = {1v1 + 2v2 + + rvr} generated by forming alllinear combinations of vectors from S is called the space spanned byS.If V is a vector space such that V = span(S), we say S is a spanningset for V. In other words, S spans V whenever each vector in V is alinear combination of vectors from S.
Shahid Hussain () Vector Spaces March 16, 2009 15 / 17
Spaces and Subspaces Subspaces
Examples
S ={(
11
),
(22
)}spans the line y = x in R2.
The unit vectors
e1 =100
, e2 =010
, e3 =001
spans R3.The unit vectors {e1, e2, . . . , en} span Rn.The finite set {1, x, x2, . . . , xn} spans the space of all polynomialssuch that deg p(x) n, and the infinite set {1, x, x2, . . .} spans thespace of all polynomials.
Shahid Hussain () Vector Spaces March 16, 2009 16 / 17
Spaces and Subspaces Subspaces
Examples
S ={(
11
),
(22
)}spans the line y = x in R2.
The unit vectors
e1 =100
, e2 =010
, e3 =001
spans R3.
The unit vectors {e1, e2, . . . , en} span Rn.The finite set {1, x, x2, . . . , xn} spans the space of all polynomialssuch that deg p(x) n, and the infinite set {1, x, x2, . . .} spans thespace of all polynomials.
Shahid Hussain () Vector Spaces March 16, 2009 16 / 17
Spaces and Subspaces Subspaces
Examples
S ={(
11
),
(22
)}spans the line y = x in R2.
The unit vectors
e1 =100
, e2 =010
, e3 =001
spans R3.The unit vectors {e1, e2, . . . , en} span Rn.
The finite set {1, x, x2, . . . , xn} spans the space of all polynomialssuch that deg p(x) n, and the infinite set {1, x, x2, . . .} spans thespace of all polynomials.
Shahid Hussain () Vector Spaces March 16, 2009 16 / 17
Spaces and Subspaces Subspaces
Examples
S ={(
11
),
(22
)}spans the line y = x in R2.
The unit vectors
e1 =100
, e2 =010
, e3 =001
spans R3.The unit vectors {e1, e2, . . . , en} span Rn.The finite set {1, x, x2, . . . , xn} spans the space of all polynomialssuch that deg p(x) n, and the infinite set {1, x, x2, . . .} spans thespace of all polynomials.
Shahid Hussain () Vector Spaces March 16, 2009 16 / 17
Spaces and Subspaces Subspaces
Problem
For a set of vectors S = {a1,a2, . . . ,an} from a subspace V Rm1, letA be the matrix containing the ais as its columns. Explain why S spansV if and only if for each b V there corresponds a column x such thatAx = b (i.e., if and only if Ax = b is a consistent system for everyb V).
Shahid Hussain () Vector Spaces March 16, 2009 17 / 17
Spaces and SubspacesSpacesSubspaces