Vector Spaces

Linear spaces are a generalization of \(d\)-dimensional Euclidean spaces (\(0\leq d\leq 3\)), etc. The relationship between related concepts can be referred to in Relationship between Euclidean Space and Linear Space.

Prerequisites: Abelian groups, fields.

Informally, a set that is closed under an operation, satisfies associativity, identity element, and inverse element forms a group. If it also satisfies commutativity, it forms an Abelian group.

If a set is closed under the four arithmetic operations, it forms a field. For related definitions, see Basic Concepts of Abstract Algebra.

Definition

A linear space (vector space) is a fundamental concept and an important object of study in linear algebra. A linear space is a module-like algebraic structure consisting of a vector set \(V\), a field \(\Bbb{P}\), an addition operation \(+\), and scalar multiplication.

Specifically, let \((V,+)\) be an Abelian group, and \(\Bbb{P}\) be a field.

Define an algebraic operation between numbers in \(\Bbb{P}\) and elements in \(V\), called scalar multiplication: \(\cdot:\Bbb{P}\times V\mapsto V\), denoted as \(p\cdot v\) or \(pv\), where \(p\) is in the field \(\Bbb{P}\), and \(v\) is in the Abelian group \(V\). This scalar multiplication is required to be closed, meaning the result of the operation is always meaningful and also lies in the group \(V\).

Moreover, it must satisfy the following conditions:

Distributivity of scalar multiplication over vector addition: For \(\mathbf u,\mathbf v\in V,a\in \Bbb{P}\), \(a(\mathbf u+\mathbf v)=a\mathbf u+a\mathbf v\)
Distributivity of scalar multiplication over scalar addition: For \(a,b\in \Bbb{P},\mathbf u\in V\), \((a+b)\mathbf u=a\mathbf u+b\mathbf u\)
Associativity of scalar multiplication (consistent with field multiplication): For \(a,b\in \Bbb{P},\mathbf u\in V\), \(a(b\mathbf u)=(ab)\mathbf u\)
Identity element of scalar multiplication: Let \(1\in \Bbb{P}\) be the multiplicative identity of \(\Bbb{P}\), then for \(u\in V\), \(1\mathbf u=\mathbf u\)

Then the algebraic system \((V,+,\cdot,\mathbb{P})\) is said to be a linear space over \(\Bbb{P}\) with respect to \(+,\cdot\), \(\Bbb{P}\) is called the base field of the linear space, elements in \(V\) are called vectors, and elements in \(\Bbb{P}\) are called scalars. When the field \(\Bbb{P}\) is the real number field, it is called a real linear space. When the field \(\Bbb{P}\) is the complex number field, it is called a complex linear space.

Whether it's a column of numbers, arrows, or something else, as long as it satisfies the above axioms, it can be considered a vector and thus can be studied using the theory of linear algebra.

The zero element in the additive group is called the zero vector, denoted as \(\mathbf 0\) or \(\mathbf\theta\).

The addition and subtraction of vectors in the original Abelian group, together with the newly defined scalar multiplication in the linear space, are collectively called linear operations.

Note

For convenience in writing, in the following:

Elements of \(V\) are not bolded.
The algebraic system \((V,+,\cdot,\mathbb{P})\) satisfying the definition of a linear space is also referred to as a linear space.

Please distinguish accordingly.

Intuitive Understanding

Loosely speaking, scalar multiplication corresponds to a kind of "scaling", elements of the base field \(\Bbb{P}\) represent the "ratio" of scaling, and vector addition corresponds to "superposition". Meanwhile, elements of \(\Bbb{P}\) also represent the range of "coordinates" for vectors.

Conditions 1-4 describe the relationship between "scaling" and "superposition". This can be understood in conjunction with arrows in a two-dimensional plane.

Simple Properties

Note

The following properties can be found in group theory, etc.

For a linear space \((V,+,\cdot,\Bbb{P})\),

\(\theta\) is unique.
For all \(\alpha\in V\), \(-\alpha\) is unique.
There exists \(0\in\mathbb{P}\) such that for all \(\alpha\in V\), \(0\alpha=\theta\).
For all \(k\in\mathbb{P}\), \(k\theta=\theta\).
\((-1)\alpha=-\alpha\), for all \(\alpha\in V\).
No zero divisors: For all \(\alpha\in V,k\in\mathbb{P}\), \(k\alpha=\theta\implies k=0\lor\alpha=\theta\).
Cancellation law for addition: For all \(\alpha,\beta,\gamma\in V\), \(\alpha+\beta=\alpha+\gamma\implies\beta=\gamma\).

In fact, the cancellation law for addition is a property of Abelian groups.

Examples

\(\Bbb{P}^n\) with addition and multiplication over the number field \(\Bbb{P}\) forms a linear space over \(\Bbb{P}\). For example, \(\Bbb{P}\) can be \(\Bbb{R}\), \(\Bbb{C}\), \(\Bbb{N}_p\) (where \(p\) is a prime), etc.
The set of \(n\times m\) matrices \(\Bbb{P}^{n\times m}\) over the number field \(\Bbb{P}\), with matrix addition and scalar multiplication, forms a linear space over \(\Bbb{P}\).
The polynomial ring \(\Bbb{P}[x]\) in one variable over the number field \(\Bbb{P}\), with polynomial addition and scalar multiplication, forms a linear space over \(\Bbb{P}\).
The set of all continuous functions on the interval \([a,b]\) (denoted \(C[a,b]\)), with "function addition" and "scalar multiplication of values with continuous functions", forms a linear space over the range.

Linear Dependence, Linear Independence

For a linear space \((V,+,\cdot,\Bbb{P})\):

\(a_1,a_2,\dots,a_n\in V\) is called a vector group of \(V\).
For \(k_1,k_2,\dots,k_n\in\Bbb{P}\), \(\sum_{i=1}^nk_ia_i\) is called a linear combination of the vector group \(a_1,a_2,\dots,a_n\).
If a vector \(\beta\in V\) can be expressed as a linear combination of the vector group \(a_1,a_2,\dots,a_n\), then \(\beta\) is said to be linearly represented by the vector group \(a_1,a_2,\dots,a_n\).
For \(k_1,k_2,\dots,k_n\in\Bbb{P}\), if the vector group \(a_1,a_2,\dots,a_n\) satisfies \(\sum_{i=1}^nk_ia_i=\theta\iff k_i=0, i=1,2,\dots,n\), then the vector group \(a_1,a_2,\dots,a_n\) is said to be linearly independent; otherwise, it is said to be linearly dependent.

It is stipulated that the zero vector is linearly dependent with any vector.

Expressions for linear representation or linear dependence can be written in the form of matrix multiplication:

\[ \beta=k_1a_1+k_2a_2+\cdots+k_ra_r=(a_1,a_2,\cdots,a_r)\begin{pmatrix} k_1 \\ k_2 \\ \vdots \\ k_r \end{pmatrix} \]

By convention, the vectors \(a\) are written side by side on the left; the scalars \(k\) are written vertically on the right, forming a "column vector".

Note: The "column vector" formed by scalars here is merely a convenient notational form and is not in the space \(V\), fundamentally different from the vectors on the left. If the vectors on the left happen to be column vectors, placing them side by side can formally constitute a "matrix", and the above product is exactly the common form of "matrix left-multiplied by a column vector".

It is pointed out below that linear representation here is also equivalent to the vector \(\beta\) lying in the image space of the matrix \((a_1,a_2\cdots,a_r)\).

According to the definition below, the zero vector will always lie in the image space. From the perspective of linear transformations, linear dependence is equivalent to multiple vectors transforming to the zero vector after transformation, while linear independence is equivalent to only the zero vector itself transforming to the zero vector.

Properties

For a linear space \((V,+,\cdot,\Bbb{P})\),

If a part of a vector group is linearly dependent, then the vector group is linearly dependent. If a vector group is linearly independent, then any non-empty part of it is linearly independent. Briefly: "Large independent, small independent"; "Small dependent, large dependent".
A vector group containing \(\theta\) is linearly dependent.
A vector group is linearly dependent if and only if some vector in the group can be linearly represented by the remaining vectors.
If a vector \(\beta\) can be linearly represented by the vector group \(a_1,a_2,\dots,a_n\), then the representation is unique if and only if the vector group \(a_1,a_2,\dots,a_n\) is linearly independent.
If the vector group \(a_1,a_2,\dots,a_n\) is linearly independent, then the vector \(\beta\) can be linearly represented by the vector group \(a_1,a_2,\dots,a_n\) if and only if the vector group \(a_1,a_2,\dots,a_n,\beta\) is linearly dependent.

Maximal Linearly Independent Set, Rank

Linear dependence can be understood as "redundant", indicating that some vectors within the group can be represented by other vectors and can be removed. After removal, what remains is a maximal linearly independent set.

For a linear space \((V,+,\cdot,\Bbb{P})\):

For a vector group \(b_1,b_2,\dots,b_m\), let \(\{a_1,a_2,\dots,a_n\}\subseteq\{b_1,b_2,\dots,b_m\}\), if:
- The vector group \(a_1,a_2,\dots,a_n\) is linearly independent.
- For all \(\beta\in\{b_1,b_2,\dots,b_m\}\setminus\{a_1,a_2,\dots,a_n\}\), the vector group \(a_1,a_2,\dots,a_n,\beta\) is linearly dependent. Then the vector group \(a_1,a_2,\dots,a_n\) is called a maximal linearly independent set of the vector group \(b_1,b_2,\dots,b_m\). Similarly, a maximal linearly independent set of the linear space \(V\) can be defined. It is stipulated that the maximal linearly independent set of the vector group \(\theta,\theta,\dots,\theta\) is the empty set, so the vector group corresponding to an all-zero matrix has no maximal linearly independent set. The method of deleting vectors from a vector group is not unique, so maximal linearly independent sets are also not unique. Conventionally, deletion is done from left to right in order. Coincidentally, deleting in order leaves vectors that exactly correspond to the columns containing element \(1\) in the row-reduced echelon form matrix remaining after Gaussian elimination, from the "row-wise" perspective. The size of a maximal linearly independent set of the vector group \(b_1,b_2,\dots,b_m\) is called the rank of the vector group, denoted \(\operatorname{rank}\{b_1,b_2,\dots,b_m\}\). It is stipulated that \(\operatorname{rank}\{\theta,\theta,\dots,\theta\}=0\). Thus, the definition of the rank of a vector group is completely consistent with the definition of the rank of a matrix.
If the vector group \(a_1,a_2,\dots,a_n\) can linearly represent all vectors in the vector group \(b_1,b_2,\dots,b_m\), then the vector group \(b_1,b_2,\dots,b_m\) is said to be linearly represented by the vector group \(a_1,a_2,\dots,a_n\).
If the vector group \(a_1,a_2,\dots,a_n\) can be linearly represented by the vector group \(b_1,b_2,\dots,b_m\), and the vector group \(b_1,b_2,\dots,b_m\) can be linearly represented by the vector group \(a_1,a_2,\dots,a_n\), then the two vector groups are said to be equivalent, denoted \(\{a_1,a_2,\dots,a_n\}\cong\{b_1,b_2,\dots,b_m\}\). Equivalence of vector groups means they span the same space. Vector groups spanning the same space are equivalent; those spanning different spaces are not equivalent. Equivalence of vector groups is a stronger condition than matrix equivalence, requiring not only the same rank but also exactly the same space. Therefore, when two matrices are concatenated horizontally, the rank must not change. Matrix equivalence only requires the same rank, so matrix equivalence means the former matrix or space can reach the latter matrix or space through an invertible transformation.

Properties

For a linear space \((V,+,\cdot,\Bbb{P})\),

Let the vector group \(a_1,a_2,\dots,a_n\) be linearly represented by the vector group \(b_1,b_2,\dots,b_m\).
- If \(n>m\), then the vector group \(a_1,a_2,\dots,a_n\) is linearly dependent.
- If the vector group \(a_1,a_2,\dots,a_n\) is linearly independent, then \(n\leq m\).
Equivalent linearly independent vector groups have the same size. Any maximal linearly independent set of a vector group has the same size.
A vector group is linearly independent if and only if its rank equals its size.
If the vector group \(a_1,a_2,\dots,a_n\) can be linearly represented by the vector group \(b_1,b_2,\dots,b_m\), then \(\operatorname{rank}\{a_1,a_2,\dots,a_n\}\leq\operatorname{rank}\{b_1,b_2,\dots,b_m\}\).
Equivalent vector groups have the same rank.

Linear Span

For a linear space \((V,+,\cdot,\Bbb{P})\), \(\left\{v=\sum_{i=1}^nk_ia_i:a_i\in V,k_i\in\Bbb{P},i=1,2,\dots,n\right\}\) also forms a linear space, called the linear space spanned by the vector group \(a_1,a_2,\dots,a_n\) (or linear span), denoted \(\operatorname{span}\{a_1,a_2,\dots,a_n\}\).

Here, the \(n\) vectors \(a\) are not necessarily linearly independent.

Linear Subspace

For a linear space \((V,+,\cdot,\Bbb{P})\), if the algebraic system \((V_1,+,\cdot,\Bbb{P})\) satisfies:

\(\varnothing\ne V_1\)
\(V_1\subseteq V\)
\(V_1\) is a linear space over \(\mathbb{P}\) with respect to \(+,\cdot\)

Then \(V_1\) is called a linear subspace of \(V\), abbreviated as subspace, denoted \(V_1\leq V\).

Any space \(V\) has two trivial subspaces: itself \(V\) and the zero subspace. The zero subspace contains only the zero vector and no linearly independent vectors.

If \(\subseteq\) in condition 2 is replaced by \(\subset\), then \(V_1\) is called a proper linear subspace of \(V\), denoted \(V_1<V\).

It is not difficult to prove: A non-empty subset \(V_1\) of a linear space \(V\) is a linear subspace if and only if linear operations are closed on \(V_1\), i.e.:

For all \(u,v\in V_1\), \(u+v\in V_1\)
For all \(v\in V_1\), for all \(k\in \Bbb{P}\), \(kv\in V_1\)

Intersection, Sum, Direct Sum, Direct Product

For linear spaces \((V_1,+,\cdot,\Bbb{P})\) and \((V_2,+,\cdot,\Bbb{P})\):

It is not difficult to verify: Addition and scalar multiplication are closed on \(V_1\cap V_2\), so \(V_1\cap V_2\) can be called the intersection of linear spaces \(V_1\) and \(V_2\). Similarly, the intersection of multiple linear spaces \(\bigcap_{i=1}^m V_i\) can be defined.
If a linear space \(V\) satisfies \(V=\{u+v|u\in V_1,v\in V_2\}\), then \(V\) is called the sum of linear spaces \(V_1\) and \(V_2\), denoted \(V=V_1+V_2\). It can be verified: \(V_1+V_2\) is the smallest subspace containing \(V_1\cup V_2\). Similarly, the sum of multiple linear spaces \(\sum_{i=1}^m V_i\) can be defined.
Let \(V=V_1+V_2\). If for any element \(v\) in the linear space \(V\), there exists a unique pair of vectors \(v_1,v_2\) such that \(v=v_1+v_2\), then \(V\) is called the direct sum of linear spaces \(V_1\) and \(V_2\), denoted \(V_1\oplus V_2\). Similarly, the direct sum of multiple linear spaces \(\bigoplus_{i=1}^m V_i\) can be defined.
The direct product \(V_1\times V_2\) of \(V_1\) and \(V_2\) is defined as the Cartesian product of the two, forming a linear space over \(\Bbb{P}\) with respect to the following addition and scalar multiplication:
1. \(+:(V_1\times V_2)\times(V_1\times V_2)\mapsto V_1\times V_2; ((u_1,v_1),(u_2,v_2))\to (u_1+u_2,v_1+v_2)\)
2. \(\cdot:\Bbb{P}\times(V_1\times V_2)\mapsto V_1\times V_2; (k,(u,v))\to (ku,kv)\) Similarly, the direct product of multiple linear spaces \(\prod_{i=1}^m V_i\) can be defined.

Examples

For the linear space \(V=\Bbb{R}^3\), define linear spaces:

\(V_1:=\{(x,0,0)|x\in\Bbb{R}\}\)
\(V_2:=\{(x,y,0)|x,y\in\Bbb{R}\}\)
\(V_3:=\{(0,y,z)|y,z\in\Bbb{R}\}\)
\(V_4:=\{(x,0,z)|x,z\in\Bbb{R}\}\)

Then

\(V_1<V_2<V\), \(V_3<V\)
\(V_2=V_1+V_2\)
\(V=V_1\oplus V_3=V_2+V_3\)
\(V_2\oplus V_3=V_4\), \(V_2\oplus V_4=V_3\), \(V_3\oplus V_4=V_2\)
\(V_2+V_3\leq V\)

Properties

Let \(V_1,V_2,V_3\) be linear spaces over \(\Bbb{P}\). Like the intersection of sets, the intersection of linear spaces obeys the following laws:
1. Commutativity: \(V_1\cap V_2=V_2\cap V_1\)
2. Associativity: \(V_1\cap(V_2\cap V_3)=(V_1\cap V_2)\cap V_3\)
Let \(V_1,V_2,V_3\) be linear spaces over \(\Bbb{P}\). Similar to the union of sets, the sum of linear spaces obeys the following laws:
1. Commutativity: \(V_1+V_2=V_2+V_1\)
2. Associativity: \(V_1+(V_2+V_3)=(V_1+V_2)+V_3\)
Let \(V_1,V_2,V_3\) be linear spaces over \(\Bbb{P}\). The intersection and sum of linear spaces have the following relationships:
1. \(V_1\cap (V_2+V_3)\supseteq (V_1\cap V_2)+(V_1\cap V_3)\)
2. \(V_1+(V_2\cap V_3)\subseteq (V_1+V_2)\cap (V_1+V_3)\)
\(\operatorname{span}\{a_1,a_2,\dots,a_n\}+\operatorname{span}\{b_1,b_2,\dots,b_m\}=\operatorname{span}\{a_1,a_2,\dots,a_n,b_1,b_2,\dots,b_m\}\)
Let \(V_1,V_2\) be linear spaces over \(\Bbb{P}\). Then the following are equivalent:
1. \(V_1+V_2=V_1\oplus V_2\)
2. There exists \(\beta\in V_1+V_2\) such that the decomposition into a sum of vectors from \(V_1\) and \(V_2\) is unique (arbitrary \(\to\) existence).
3. The decomposition of \(\theta\) into a sum of vectors from \(V_1\) and \(V_2\) is unique.
4. \(V_1\cap V_2=\{\theta\}\)
Proof

\(1\implies 2\): Follows directly from the definition.

\(2 \implies 3\):

Let \(\beta=\beta_1+\beta_2\), where \(\beta_1\in V_1, \beta_2\in V_2\). If \(\theta=\alpha_1+\alpha_2\), \(\theta\ne\alpha_1\in V_1,\alpha_2\in V_2\), then \(\beta=\beta+\theta=(\beta_1+\alpha_1)+(\beta_2+\alpha_2)\).

But \(\beta_1\ne\beta_1+\alpha_1\), contradicting the condition.

\(3 \implies 4\):

Take a non-zero vector \(\alpha\) from \(V_1\) and \(V_2\), then \(\theta=\alpha+(-\alpha)=(-\alpha)+\alpha\), contradicting the condition.

\(4 \implies 1\):

If \(V_1+V_2\) is not a direct sum, then there exists \(\beta\in V_1+V_2\) such that \(\beta=\beta_1+\beta_2=\gamma_1+\gamma_2\), where \(\beta_1,\gamma_1\in V_1,\beta_2,\gamma_2\in V_2\) and \(\beta_1,\beta_2,\gamma_1,\gamma_2\) are all distinct.

Then \(\theta\ne\beta_1-\gamma_1=\gamma_2-\beta_2\in V_1\cap V_2\), contradicting the condition.

Isomorphism

Let \(V,V'\) be linear spaces over a field \(\Bbb{P}\). If there exists a bijection \(\sigma:V\mapsto V'\) that preserves addition and scalar multiplication, i.e., for all \(u,v\in V\), for all \(k\in\Bbb{P}\):

\(\sigma(u+v)=\sigma(u)+\sigma(v)\)
\(\sigma(ku)=k\sigma(u)\)

Then \(\sigma\) is called an isomorphism from \(V\) to \(V'\), and \(V\) and \(V'\) are said to be isomorphic, denoted \(V\cong V'\).

Note

If \(\sigma\) is injective, it can be called a monomorphism; if \(\sigma\) is surjective, it can be called an epimorphism.

Properties

Two linear spaces over a field \(\Bbb{P}\) are isomorphic if and only if they have the same dimension. (For the definition of dimension, see Basis.)
(Corollary of 1) An \(n\)-dimensional linear space over a field \(\Bbb{P}\) is isomorphic to the linear space \(\Bbb{P}^n\).

Note

This property essentially allows us to treat coordinates and vectors as equivalent.

Relationship between Euclidean Space and Linear Space

Taking the familiar three-dimensional Euclidean space as an example, the corresponding relationships of some related concepts in linear space are shown in the following table:

Three-dimensional Euclidean Space	Linear Space
Vector	Vector
Perpendicular	Orthogonal (i.e., inner product is \(0\))
Three vectors collinear/coplanar	\(k\) vectors linearly dependent
Three vectors non-coplanar	\(k\) vectors linearly independent
Basis vectors	Basis
Dimension of space	Dimension of space

Applications

From this section onward, we mainly discuss the "column-wise" perspective of linear equation systems.

Matrix \(A\) itself is also composed of column vectors. Viewing \(A\) itself as a column vector group, and \(x\) as unknown coefficients, we consider whether the column vectors in \(A\) can be combined with the unknowns to form the column vector \(b\). Here, the column vector \(x\) is completely unknown.

The equation \(Ax=b\) is rearranged as:

\[ \alpha_1 x_1 +\alpha_2 x_2 +\cdots+\alpha_n x_n=b \]

Here, in matrix multiplication, the matrix \(A\) on the left can be viewed as a vector group, i.e., a set of column vectors. This set of column vectors serves as a basis, spanning a space, and we investigate whether the column vector \(b\) lies in this space.

Column-wise View of Solutions to Linear Equations

Rank is the number of vectors in a maximal linearly independent set, representing "constraints". The remaining vectors will give degrees of freedom to the solution, i.e., allow redundant vectors in other directions.

If we denote \(n\) as the number of columns of matrix \(A\), i.e., the number of column vectors, and \(r(A)\) as the rank of matrix \(A\), then the degrees of freedom \(S\) are:

\[ S=n-r(A) \]

The set of all solutions to the equation system also forms a vector group. The degrees of freedom \(S\) is the rank of the solution vector group for \(Ax=0\), i.e., the dimension of the kernel space discussed below.

Equivalent Equation Systems

The common solutions of two equation systems are defined as the intersection of their solution sets.

Two equation systems are equivalent if their solution sets are equal. Equation systems with equal solution sets are equivalent; those with unequal solution sets are not equivalent.

Equivalence of equation systems is a stronger condition than matrix equivalence, requiring not only the same rank but also that the rank remains unchanged when the two matrices are concatenated vertically.

Comparing with vector group equivalence, vector group equivalence requires horizontal concatenation of matrices without changing the rank. Therefore, we have the following relationship:

Matrix equivalence does not necessarily imply corresponding vector group equivalence or equation system equivalence, but if vector groups are equivalent or equation systems are equivalent, the corresponding matrices must be equivalent (same rank).

If the vector groups corresponding to matrices are equivalent, then after transposing the matrices, the corresponding equation systems are equivalent, and vice versa.

Kernel Space and Image Space of a Matrix

The kernel space and image space discussed here are from the perspective of linear spaces.

For a matrix \(A\), let \(W\) be the set of all solutions \(x\) to the equation \(Ax=0\). Then \(W\) is a linear space, and the scalar field of \(W\) is the same as the field of the elements of \(A\).

\(W\) is then called the kernel space of matrix \(A\), denoted \(N(A)\).

The kernel space \(N(A)\) of matrix \(A\) is the solution space of the equation \(Ax=0\). According to the definition of basis later, the fundamental system of solutions of this equation is a basis of the kernel space.

If matrix \(A\) is invertible, then the kernel space \(N(A)\) of \(A\) contains only the zero vector.

For matrix \(A\), its \(n\) columns are vectors \(\alpha\). The space spanned by these \(n\) column vectors \(\alpha\) is called the image space of \(A\), or column space, denoted:

\[ R(A)=\operatorname{span}\{\alpha_1,\alpha_2,\cdots,\alpha_n\} \]

According to the definition of dimension later, the dimension of the image space equals the rank of matrix \(A\).

By definition, for every element \(y\) in the image space \(R(A)\), there is a corresponding representation:

\[ y=k_1\alpha_1+k_2\alpha_2+\cdots+k_n\alpha_n=(\alpha_1,\alpha_2,\cdots,\alpha_n)\begin{pmatrix}k_1\\k_2\\\vdots\\k_n\end{pmatrix}=A\begin{pmatrix}k_1\\k_2\\\vdots\\k_n\end{pmatrix} \]

Therefore, the image space \(R(A)\) is the range of \(Ax\) for any vector \(x\).

Similarly, the row space of \(A\) can be defined as the range of \(A\)'s transpose, \(R(A^T)\).

Since the row rank of a matrix equals its column rank, the dimension of the row space is also the rank of the matrix. Thus, transposition changes the image space but not its dimension.

Here we can establish correspondences with the previous discussion:

Vector group equivalence is equivalent to having the same image space \(R(A)\) for the corresponding matrices.

Equation system equivalence is equivalent to having the same row space \(R(A^T)\) for the corresponding matrices.

References and Notes

Qiu Weisheng, Advanced Algebra (Part 2). Tsinghua University Press.
Vector space.Wikipedia, The Free Encyclopedia.

本页面最近更新：，更新历史
发现错误？想一起完善？在 GitHub 上编辑此页！
本页面贡献者：OI-wiki
本页面的全部内容在 CC BY-SA 4.0 和 SATA 协议之条款下提供，附加条款亦可能应用

Vector Spaces

Definition

Intuitive Understanding

Simple Properties

Examples

Related Concepts

Linear Dependence, Linear Independence

Properties

Maximal Linearly Independent Set, Rank

Properties

Linear Span

Linear Subspace

Intersection, Sum, Direct Sum, Direct Product

Examples

Properties

Isomorphism

Properties

Relationship between Euclidean Space and Linear Space

Applications

Column-wise View of Solutions to Linear Equations

Equivalent Equation Systems

Kernel Space and Image Space of a Matrix

References and Notes