The Central Idea
A vector space contains two kinds of mathematical objects:
- vectors, which belong to the vector space;
- scalars, which belong to a field.
Vectors can be added to one another, and vectors can be multiplied by scalars.
These two operations satisfy a collection of axioms that make linear algebra possible.
What Is a Scalar?
A scalar is an element of the field used to scale vectors.
Common scalar fields are
\[ \mathbb{R} \]and
\[ \mathbb{C}. \]A scalar may stretch, shrink, reverse, or otherwise scale a vector.
What Is a Vector?
A vector is an element of a vector space.
In elementary geometry, vectors are often represented as directed arrows.
In algebra, however, a vector can be any object that participates in vector addition and scalar multiplication according to the vector-space axioms.
Vectors Are More General Than Arrows
Examples of vectors include:
- ordered pairs such as \((2,-1)\);
- ordered triples such as \((1,0,4)\);
- \(n\)-tuples in \(\mathbb{R}^n\);
- matrices of a fixed size;
- polynomials;
- functions;
- sequences.
The word vector therefore describes structural behavior, not just geometric appearance.
Coordinate Vectors
A vector in
\[ \mathbb{R}^2 \]has the form
\[ \mathbf{v}=(v_1,v_2). \]A vector in
\[ \mathbb{R}^3 \]has the form
\[ \mathbf{v}=(v_1,v_2,v_3). \]More generally,
\[ \mathbb{R}^n \]consists of all ordered \(n\)-tuples of real numbers.
Vector Addition
In coordinate spaces, vectors are added component by component.
If
\[ \mathbf{u}=(u_1,u_2) \]and
\[ \mathbf{v}=(v_1,v_2), \]then
\[ \boxed{ \mathbf{u}+\mathbf{v} = (u_1+v_1,\;u_2+v_2) }. \]Example: Vector Addition
Let
\[ \mathbf{u}=(2,3) \]and
\[ \mathbf{v}=(-1,4). \]Then
\[ \mathbf{u}+\mathbf{v} = (2+(-1),\;3+4) \]so
\[ \boxed{ \mathbf{u}+\mathbf{v}=(1,7) }. \]Scalar Multiplication
A scalar multiplies every component of a coordinate vector.
If
\[ c\in\mathbb{R} \]and
\[ \mathbf{v}=(v_1,v_2), \]then
\[ \boxed{ c\mathbf{v} = (cv_1,\;cv_2) }. \]Example: Scalar Multiplication
Let
\[ \mathbf{v}=(3,-2) \]and
\[ c=4. \]Then
\[ 4\mathbf{v} = (12,-8). \]Therefore,
\[ \boxed{ 4(3,-2)=(12,-8) }. \]Scalar Multiplication Is an External Operation
Vector addition has the form
\[ V\times V\to V. \]Scalar multiplication has the form
\[ \boxed{ F\times V\to V }. \]Here \(F\) is the scalar field and \(V\) is the vector space.
Because the two inputs come from different sets, scalar multiplication is not a binary operation on \(V\) in the strict internal sense.
What Is a Vector Space?
Let \(F\) be a field.
A vector space over \(F\) is a set \(V\) equipped with vector addition and scalar multiplication satisfying the vector-space axioms.
We may write
\[ \boxed{ V\text{ is a vector space over }F }. \]The Scalar Field Matters
A vector space is always defined relative to a field of scalars.
For example,
\[ \mathbb{R}^2 \]is naturally a vector space over
\[ \mathbb{R}. \]The same underlying set might behave differently if a different scalar field is chosen.
The Vector-Space Axioms
For vectors
\[ \mathbf{u},\mathbf{v},\mathbf{w}\in V \]and scalars
\[ a,b\in F, \]a vector space satisfies the following laws.
Closure Under Vector Addition
If
\[ \mathbf{u},\mathbf{v}\in V, \]then
\[ \boxed{ \mathbf{u}+\mathbf{v}\in V }. \]Adding two vectors must produce another vector in the same space.
Commutativity of Vector Addition
Vector addition satisfies
\[ \boxed{ \mathbf{u}+\mathbf{v} = \mathbf{v}+\mathbf{u} }. \]Therefore the additive operation is commutative.
Associativity of Vector Addition
Vector addition satisfies
\[ \boxed{ (\mathbf{u}+\mathbf{v})+\mathbf{w} = \mathbf{u}+(\mathbf{v}+\mathbf{w}) }. \]The Zero Vector
Every vector space contains an additive identity called the zero vector.
It is written
\[ \boxed{\mathbf{0}}. \]It satisfies
\[ \mathbf{v}+\mathbf{0} = \mathbf{0}+\mathbf{v} = \mathbf{v}. \]Zero Vectors Depend on the Space
In
\[ \mathbb{R}^2, \]the zero vector is
\[ (0,0). \]In
\[ \mathbb{R}^3, \]it is
\[ (0,0,0). \]In a polynomial space, the zero vector is the zero polynomial.
In a function space, it is the function that outputs \(0\) everywhere.
Additive Inverses of Vectors
Every vector
\[ \mathbf{v}\in V \]has an additive inverse
\[ -\mathbf{v} \]satisfying
\[ \boxed{ \mathbf{v}+(-\mathbf{v})=\mathbf{0} }. \]Closure Under Scalar Multiplication
If
\[ a\in F \]and
\[ \mathbf{v}\in V, \]then
\[ \boxed{ a\mathbf{v}\in V }. \]Scaling a vector must keep the result inside the vector space.
Distributivity Over Vector Addition
Scalar multiplication distributes over vector addition:
\[ \boxed{ a(\mathbf{u}+\mathbf{v}) = a\mathbf{u}+a\mathbf{v} }. \]Distributivity Over Scalar Addition
Scalar addition also distributes through scalar multiplication:
\[ \boxed{ (a+b)\mathbf{v} = a\mathbf{v}+b\mathbf{v} }. \]Compatibility of Scalar Multiplication
Scalar multiplication satisfies
\[ \boxed{ a(b\mathbf{v}) = (ab)\mathbf{v} }. \]Scaling first by \(b\) and then by \(a\) is the same as scaling once by the product \(ab\).
Scalar Identity
The multiplicative identity
\[ 1\in F \]satisfies
\[ \boxed{ 1\mathbf{v}=\mathbf{v} }. \]Thus multiplying by scalar \(1\) leaves every vector unchanged.
Vector Addition Forms an Abelian Group
The additive axioms imply that
\[ \boxed{(V,+)} \]is an abelian group.
This connects vector spaces directly to the group structures studied earlier.
Scalar Multiplication Connects the Field to the Vector Space
The field \(F\) supplies the scalars.
The vector space \(V\) supplies the vectors.
Scalar multiplication
\[ F\times V\to V \]links the two algebraic structures.
The distributive and compatibility laws ensure that field arithmetic and vector addition interact consistently.
Example: \(\mathbb{R}^2\) Is a Vector Space
Consider
\[ \mathbb{R}^2 = \{(x,y):x,y\in\mathbb{R}\}. \]Under ordinary coordinate addition and real scalar multiplication, all vector-space axioms hold.
Therefore,
\[ \boxed{ \mathbb{R}^2 \text{ is a vector space over }\mathbb{R} }. \]Example: \(\mathbb{R}^n\)
More generally,
\[ \mathbb{R}^n \]is a vector space over \(\mathbb{R}\).
Addition and scalar multiplication are performed component by component:
\[ (u_1,\ldots,u_n)+(v_1,\ldots,v_n) = (u_1+v_1,\ldots,u_n+v_n), \]and
\[ c(v_1,\ldots,v_n) = (cv_1,\ldots,cv_n). \]Example: Complex Coordinate Spaces
The set
\[ \mathbb{C}^n \]is a vector space over the field
\[ \mathbb{C}. \]Its vectors are ordered \(n\)-tuples of complex numbers, and its scalars are complex numbers.
Example: Matrices as Vectors
Let
\[ M_{m\times n}(\mathbb{R}) \]be the set of all \(m\times n\) real matrices.
Under matrix addition and real scalar multiplication, this set forms a vector space over \(\mathbb{R}\).
A matrix can therefore be treated as a vector in a suitable vector space.
Matrix Addition as Vector Addition
For matrices \(A\) and \(B\) of the same size,
\[ A+B \]is computed entry by entry.
For example,
\[ \begin{pmatrix} 1&2\\ 3&4 \end{pmatrix} + \begin{pmatrix} 5&0\\ -1&2 \end{pmatrix} = \begin{pmatrix} 6&2\\ 2&6 \end{pmatrix}. \]Scalar Multiplication of Matrices
Multiplying a matrix by a scalar multiplies every entry.
For example,
\[ 3 \begin{pmatrix} 1&-2\\ 0&4 \end{pmatrix} = \begin{pmatrix} 3&-6\\ 0&12 \end{pmatrix}. \]Example: Polynomials as Vectors
Let
\[ P_n(\mathbb{R}) \]denote the real polynomials of degree at most \(n\).
Under ordinary polynomial addition and real scalar multiplication,
\[ P_n(\mathbb{R}) \]forms a vector space over \(\mathbb{R}\).
Polynomial Addition
Let
\[ p(x)=1+2x \]and
\[ q(x)=3-x+x^2. \]Then
\[ p(x)+q(x) = 4+x+x^2. \]The result is another polynomial in the same polynomial space when the degree restriction is respected.
Scalar Multiplication of Polynomials
If
\[ p(x)=1-3x+2x^2, \]then
\[ 4p(x) = 4-12x+8x^2. \]Scalar multiplication changes the coefficients while preserving the polynomial form.
Example: Function Spaces
Consider a suitable set of real-valued functions
\[ f:X\to\mathbb{R}. \]Define addition pointwise:
\[ (f+g)(x)=f(x)+g(x), \]and scalar multiplication pointwise:
\[ (cf)(x)=c\,f(x). \]Under these operations, many collections of functions form vector spaces.
The Zero Function
In a function vector space, the zero vector is the function
\[ \mathbf{0}(x)=0 \]for every input \(x\).
It satisfies
\[ f+\mathbf{0}=f. \]Not Every Subset Is a Vector Space
A collection of vectors may fail to satisfy vector-space closure.
Consider
\[ S=\{(x,y)\in\mathbb{R}^2:x\ge0\}. \]The vector
\[ (1,0) \]belongs to \(S\), but multiplying by scalar \(-1\) gives
\[ (-1,0), \]which does not belong to \(S\).
Therefore \(S\) is not a vector space over \(\mathbb{R}\).
The Zero Vector Must Be Present
Every vector space must contain
\[ \mathbf{0}. \]Therefore any proposed vector space that omits the zero vector can be rejected immediately.
Why \(0\mathbf{v}=\mathbf{0}\)
Here the first \(0\) is the scalar zero and
\[ \mathbf{0} \]is the zero vector.
Since
\[ 0+0=0 \]in the scalar field,
\[ (0+0)\mathbf{v}=0\mathbf{v}. \]Distribute:
\[ 0\mathbf{v}+0\mathbf{v}=0\mathbf{v}. \]Add the inverse of \(0\mathbf{v}\) to both sides:
\[ \boxed{ 0\mathbf{v}=\mathbf{0} }. \]Why \(a\mathbf{0}=\mathbf{0}\)
Since
\[ \mathbf{0}+\mathbf{0}=\mathbf{0}, \]we have
\[ a(\mathbf{0}+\mathbf{0})=a\mathbf{0}. \]Distribute:
\[ a\mathbf{0}+a\mathbf{0}=a\mathbf{0}. \]Therefore,
\[ \boxed{ a\mathbf{0}=\mathbf{0} }. \]Why \((-1)\mathbf{v}=-\mathbf{v}\)
Since
\[ 1+(-1)=0, \]multiply by \(\mathbf{v}\):
\[ (1+(-1))\mathbf{v}=0\mathbf{v}. \]Distribute:
\[ \mathbf{v}+(-1)\mathbf{v}=\mathbf{0}. \]Thus \((-1)\mathbf{v}\) is the additive inverse of \(\mathbf{v}\):
\[ \boxed{ (-1)\mathbf{v}=-\mathbf{v} }. \]Subspaces Preview
A subset of a vector space that is itself a vector space under the same operations is called a subspace.
This parallels the subgroup idea studied earlier.
A practical subspace test checks:
- the zero vector is present;
- closure under vector addition;
- closure under scalar multiplication.
Example: A Line Through the Origin
Consider
\[ L=\{(x,y)\in\mathbb{R}^2:y=2x\}. \]Every vector in \(L\) has the form
\[ (t,2t) \]for some \(t\in\mathbb{R}\).
Adding two such vectors gives
\[ (s,2s)+(t,2t) = (s+t,2(s+t)), \]which remains in \(L\).
Scalar multiplication also remains in \(L\).
Therefore \(L\) is a subspace of \(\mathbb{R}^2\).
Example: A Shifted Line Is Not a Subspace
Consider
\[ L=\{(x,y)\in\mathbb{R}^2:y=2x+1\}. \]The zero vector
\[ (0,0) \]does not satisfy
\[ 0=2(0)+1. \]Therefore the line does not contain the zero vector and cannot be a subspace.
Vectors and Geometric Interpretation
In \(\mathbb{R}^2\) or \(\mathbb{R}^3\), vectors may still be visualized as arrows.
Vector addition corresponds geometrically to combining displacements.
Scalar multiplication changes magnitude and may reverse direction.
But the vector-space axioms are more general than this geometric picture.
Positive, Negative, and Zero Scalars
In a real vector space:
- if \(c>1\), the vector is stretched;
- if \(0
- if \(c<0\), direction is reversed and magnitude is scaled;
- if \(c=0\), the result is the zero vector.
These geometric interpretations are useful in coordinate spaces.
Vector Spaces Separate Structure from Coordinates
Coordinate vectors such as
\[ (1,2,3) \]are concrete examples.
But vector-space theory focuses on the laws satisfied by addition and scalar multiplication rather than the specific form of the objects.
This allows the same theory to apply to vectors, matrices, polynomials, and functions.
Why the Scalar Field Must Be a Field
Linear algebra frequently requires division by nonzero scalars.
Fields guarantee that every nonzero scalar
\[ a \]has inverse
\[ a^{-1}. \]This makes it possible to solve equations such as
\[ a\mathbf{v}=\mathbf{w} \]by multiplying by \(a^{-1}\) when \(a\ne0\).
Example: Solving a Scalar Equation
Suppose
\[ 3\mathbf{v}=(6,-9). \]Multiply by
\[ \frac13. \]Then
\[ \mathbf{v} = \frac13(6,-9) = (2,-3). \]Therefore,
\[ \boxed{ \mathbf{v}=(2,-3) }. \]Common Error: Confusing Scalars and Vectors
A scalar belongs to the field.
A vector belongs to the vector space.
In
\[ \mathbb{R}^2, \]the number
\[ 3 \]is a scalar, while
\[ (3,0) \]is a vector.
Common Error: Treating Scalar Multiplication as Vector Multiplication
The expression
\[ c\mathbf{v} \]combines a scalar and a vector.
It is not the same as multiplying two vectors together.
Dot products, cross products, and other vector products are separate operations with separate definitions.
Common Error: Assuming Every Collection of Coordinate Vectors Is a Vector Space
A subset must remain closed under both addition and scalar multiplication.
For example, the first quadrant
\[ \{(x,y):x\ge0,\ y\ge0\} \]is not a vector space over \(\mathbb{R}\), because multiplying by a negative scalar leaves the set.
Common Error: Forgetting the Zero Vector
Every vector space must contain its additive identity.
If a proposed set of vectors does not contain
\[ \mathbf{0}, \]it is not a vector space under the usual operations.
Common Error: Confusing Scalar Zero with the Zero Vector
The scalar
\[ 0 \]belongs to the field.
The zero vector
\[ \mathbf{0} \]belongs to the vector space.
They play related but distinct structural roles.
A Reliable Vector-Space Checklist
- Identify the proposed vector set \(V\).
- Identify the scalar field \(F\).
- Identify vector addition.
- Identify scalar multiplication.
- Check closure under vector addition.
- Check that the zero vector belongs to \(V\).
- Check additive inverses.
- Check closure under scalar multiplication.
- Verify the distributive and compatibility laws.
- Confirm that scalar \(1\) leaves vectors unchanged.
Example: Full Vector-Space Check
Consider
\[ V=\{(x,y)\in\mathbb{R}^2:y=3x\}. \]Every vector has the form
\[ (t,3t). \]Zero vector
Choosing \(t=0\) gives
\[ (0,0)\in V. \]Closure under addition
If
\[ \mathbf{u}=(s,3s) \]and
\[ \mathbf{v}=(t,3t), \]then
\[ \mathbf{u}+\mathbf{v} = (s+t,3s+3t) = (s+t,3(s+t)), \]which belongs to \(V\).
Closure under scalar multiplication
For \(c\in\mathbb{R}\),
\[ c(t,3t) = (ct,3ct), \]which also belongs to \(V\).
Therefore,
\[ \boxed{ V\text{ is a vector space over }\mathbb{R} }. \]Practice 1: Vector Addition
Compute
\[ (2,-1)+(4,3). \]Practice 2: Scalar Multiplication
Compute
\[ -2(3,5). \]Practice 3: Zero Vector
State the zero vector in
\[ \mathbb{R}^4. \]Practice 4: Additive Inverse
Find the additive inverse of
\[ (4,-7,2). \]Practice 5: Scalar Zero
Evaluate
\[ 0(5,-3,8). \]Practice 6: Matrix Vector Space
Is the set of all \(2\times2\) real matrices a vector space over \(\mathbb{R}\) under ordinary matrix addition and scalar multiplication?
Practice 7: Polynomial Vector Space
Is
\[ P_2(\mathbb{R}) \]a vector space over \(\mathbb{R}\)?
Practice 8: Subspace Test
Determine whether
\[ S=\{(x,y)\in\mathbb{R}^2:y=-x\} \]is a subspace of \(\mathbb{R}^2\).
Practice 9: Not a Subspace
Explain why
\[ S=\{(x,y)\in\mathbb{R}^2:y=-x+2\} \]is not a subspace.
Practice 10: Identify the Error
A student says:
“Any set of ordered pairs is automatically a vector space because ordered pairs are vectors.”
Explain the error.
Practice Answers
Practice 1
\[ (2,-1)+(4,3) = (6,2). \]Therefore,
\[ \boxed{(6,2)}. \]Practice 2
\[ -2(3,5) = (-6,-10). \]Therefore,
\[ \boxed{(-6,-10)}. \]Practice 3
The zero vector is
\[ \boxed{(0,0,0,0)}. \]Practice 4
The additive inverse is
\[ \boxed{(-4,7,-2)}. \]Practice 5
Scalar zero sends every vector to the zero vector:
\[ \boxed{ 0(5,-3,8)=(0,0,0) }. \]Practice 6
Yes. The set of all \(2\times2\) real matrices is closed under matrix addition and real scalar multiplication, contains the zero matrix, and satisfies the vector-space axioms.
Practice 7
Yes. Polynomials of degree at most \(2\) are closed under addition and scalar multiplication, and the usual operations satisfy the vector-space axioms.
Practice 8
Yes. The zero vector satisfies
\[ 0=-0. \]If
\[ (a,-a) \]and
\[ (b,-b) \]belong to the set, then
\[ (a+b,-a-b) \]also belongs to the set.
Scalar multiples remain in the set as well.
Therefore it is a subspace.
Practice 9
The zero vector
\[ (0,0) \]does not satisfy
\[ 0=-0+2. \]Therefore the set does not contain the zero vector and cannot be a subspace.
Practice 10
Being an ordered pair does not automatically guarantee vector-space structure.
The set must be closed under the specified vector addition and scalar multiplication and must satisfy all vector-space axioms.
What to Remember
A vector space \(V\) over a field \(F\) contains vectors that can be added and scaled.
Vector addition has the form
\[ V\times V\to V, \]while scalar multiplication has the form
\[ \boxed{ F\times V\to V }. \]The main structural ideas are:
- \((V,+)\) is an abelian group;
- the zero vector acts as the additive identity;
- every vector has an additive inverse;
- scalar multiplication stays inside the vector space;
- scalar multiplication distributes over vector addition and scalar addition;
- scalar multiplication is compatible with multiplication in the scalar field;
- scalar \(1\) leaves every vector unchanged.
Coordinate lists, matrices, polynomials, and functions can all be vectors because vectorhood is determined by algebraic structure rather than appearance.
Where We Go Next
Once vectors can be added and scaled, we can ask what vectors can be built from other vectors and whether some vectors are redundant.
The next lesson, Linear Combinations, Span, and Independence, develops the central ideas used to construct vector spaces from smaller collections of vectors and determine when vectors contribute genuinely new directions.