Linear Maps

3A. Vector Space of Linear Maps

1) Linear Map

Linear Map

A linear map from $V$ to $W$ is a function

$$T:V\to W$$

that satisfies additivity and homogeneity.

$$T(u+v)=T(u)+T(v)$$

and

$$T(\lambda v)=\lambda T(v).$$

Meaning: A linear map preserves vector addition and scalar multiplication.

2) Space of Linear Maps

$\mathcal{L}(V,W)$

The set of all linear maps from $V$ to $W$ is denoted by

$$\mathcal{L}(V,W).$$

Meaning: $\mathcal{L}(V,W)$ contains every linear transformation from $V$ into $W$.

$\mathcal{L}(V)$

The set of all linear maps from $V$ to itself is denoted by

$$\mathcal{L}(V)=\mathcal{L}(V,V).$$

Meaning: $\mathcal{L}(V)$ contains all linear operators on $V$.

3) Zero Map and Identity Operator

Zero Map

The zero map is defined by

$$T(v)=0$$

for every $v\in V$.

Meaning: The zero map sends every vector to the zero vector.

Identity Operator

The identity operator $I\in\mathcal{L}(V)$ is defined by

$$I(v)=v.$$

Meaning: The identity operator leaves every vector unchanged.

4) Linear Map Lemma

Linear Map Lemma

Suppose

$$v_1,\ldots,v_n$$

is a basis of $V$ and

$$w_1,\ldots,w_n\in W.$$

Then there exists a unique linear map

$$T:V\to W$$

such that

$$T(v_k)=w_k$$

for every $k=1,\ldots,n$.

Meaning: A linear map is completely determined by its values on a basis.

5) Addition and Scalar Multiplication of Linear Maps

Addition

For $S,T\in\mathcal{L}(V,W)$,

$$(S+T)(v)=S(v)+T(v).$$

Meaning: Linear maps are added by adding their outputs.

Scalar Multiplication

For $\lambda\in\mathbb{F}$,

$$(\lambda T)(v)=\lambda T(v).$$

Meaning: A linear map is multiplied by a scalar by multiplying every output by that scalar.

Vector Space of Linear Maps

With these operations,

$$\mathcal{L}(V,W)$$

is a vector space.

Meaning: Linear maps themselves can be treated as vectors.

6) Product of Linear Maps

Product of Linear Maps

If

$$T\in\mathcal{L}(U,V)$$

and

$$S\in\mathcal{L}(V,W),$$

then

$$ST\in\mathcal{L}(U,W)$$

is defined by

$$(ST)(u)=S(T(u)).$$

Meaning: The product of linear maps is ordinary function composition.

Associativity

$$(T_1T_2)T_3=T_1(T_2T_3).$$

Meaning: The grouping of composed linear maps does not matter.

Identity

$$TI=IT=T.$$

Meaning: Composing with the identity operator changes nothing.

Distributivity

$$(S_1+S_2)T=S_1T+S_2T$$

and

$$S(T_1+T_2)=ST_1+ST_2.$$

Meaning: Composition distributes over addition.

Noncommutativity

In general,

$$ST\neq TS.$$

Meaning: The order in which linear maps are applied usually matters.

7) Linear Maps Preserve Zero

Zero Preservation

Every linear map satisfies

$$T(0)=0.$$

Meaning: A linear map must send the zero vector to the zero vector.

3B. Null Spaces and Ranges

1) Null Space

Null Space

For $T\in\mathcal{L}(V,W)$,

$$\operatorname{null}T = \{v\in V\mid T(v)=0\}.$$

Meaning: The null space contains all vectors that are sent to zero.

Null Space Is a Subspace

$$\operatorname{null}T$$

is a subspace of $V$.

Meaning: Vectors mapped to zero form their own vector space inside the domain.

2) Injectivity

Injective

A linear map $T:V\to W$ is injective if

$$T(u)=T(v)$$

implies

$$u=v.$$

Meaning: Different input vectors cannot produce the same output.

Injectivity Criterion

A linear map $T$ is injective if and only if

$$\operatorname{null}T=\{0\}.$$

Meaning: A linear map is one-to-one exactly when only the zero vector is mapped to zero.

3) Range

Range

The range of $T\in\mathcal{L}(V,W)$ is

$$\operatorname{range}T = \{T(v)\mid v\in V\}.$$

Meaning: The range contains every vector that can actually be produced by $T$.

Range Is a Subspace

$$\operatorname{range}T$$

is a subspace of $W$.

Meaning: All possible outputs of a linear map form a vector space inside the target space.

4) Surjectivity

Surjective

A linear map $T:V\to W$ is surjective if

$$\operatorname{range}T=W.$$

Meaning: Every vector in the target space is produced by some vector in the domain.

5) Fundamental Theorem of Linear Maps

Fundamental Theorem of Linear Maps

If $V$ is finite-dimensional and

$$T\in\mathcal{L}(V,W),$$

then

$$\dim V = \dim\operatorname{null}T + \dim\operatorname{range}T.$$

Meaning: The dimensions lost to the null space and preserved in the range add up to the dimension of the domain.

6) Dimension Restrictions

Injectivity Dimension Condition

If $V$ and $W$ are finite-dimensional and

$$\dim V>\dim W,$$

then no linear map from $V$ to $W$ can be injective.

Meaning: A larger-dimensional space cannot be injected linearly into a smaller-dimensional space.

Surjectivity Dimension Condition

If

$$\dim V<\dim W,$$

then no linear map from $V$ to $W$ can be surjective.

Meaning: A smaller-dimensional space cannot linearly fill a larger-dimensional space.

3C. Matrices

1) Matrix

Matrix

An $m$-by-$n$ matrix is a rectangular array with $m$ rows and $n$ columns.

The entry in row $j$ and column $k$ is denoted by

$$A_{j,k}.$$

Meaning: A matrix stores numbers in a rectangular arrangement.

2) Matrix of a Linear Map

Matrix of a Linear Map

Suppose

$$v_1,\ldots,v_n$$

is a basis of $V$ and

$$w_1,\ldots,w_m$$

is a basis of $W$.

For $T\in\mathcal{L}(V,W)$, write

$$T(v_k) = A_{1,k}w_1+\cdots+A_{m,k}w_m.$$

The matrix of $T$ is

$$\mathcal{M}(T) = \begin{pmatrix} A_{1,1} & \cdots & A_{1,n}\\ \vdots & & \vdots\\ A_{m,1} & \cdots & A_{m,n} \end{pmatrix}.$$

Meaning: Column $k$ contains the coordinates of $T(v_k)$ in the basis of $W$.

3) Matrix Addition and Scalar Multiplication

Matrix Addition

Matrices of the same size are added entry by entry.

$$(A+C)_{j,k}=A_{j,k}+C_{j,k}.$$

Meaning: Corresponding entries are added.

Scalar Multiplication

$$(\lambda A)_{j,k}=\lambda A_{j,k}.$$

Meaning: Every entry is multiplied by the same scalar.

Matrix of a Sum

$$\mathcal{M}(S+T) = \mathcal{M}(S)+\mathcal{M}(T).$$

Meaning: Adding linear maps corresponds to adding their matrices.

Matrix of a Scalar Multiple

$$\mathcal{M}(\lambda T) = \lambda\mathcal{M}(T).$$

Meaning: Scalar multiplication of linear maps corresponds to scalar multiplication of matrices.

4) Matrix Space

$\mathbb{F}^{m,n}$

The vector space of all $m$-by-$n$ matrices over $\mathbb{F}$ is denoted by

$$\mathbb{F}^{m,n}.$$

Its dimension is

$$\dim\mathbb{F}^{m,n}=mn.$$

Meaning: An $m$-by-$n$ matrix has $mn$ independent entries.

5) Matrix Multiplication

Matrix Multiplication

If

$$A\in\mathbb{F}^{m,n}$$

and

$$B\in\mathbb{F}^{n,p},$$

then

$$AB\in\mathbb{F}^{m,p}$$

with

$$(AB)_{j,k} = \sum_{r=1}^{n}A_{j,r}B_{r,k}.$$

Meaning: Each entry of the product is obtained by multiplying a row of $A$ by a column of $B$.

Matrix of a Product

$$\mathcal{M}(ST) = \mathcal{M}(S)\mathcal{M}(T).$$

Meaning: Composition of linear maps corresponds to matrix multiplication.

6) Matrix-Vector Multiplication

Matrix-Vector Product

If the columns of $A$ are

$$A_{\cdot,1},\ldots,A_{\cdot,n}$$

and

$$b= \begin{pmatrix} b_1\\ \vdots\\ b_n \end{pmatrix},$$

then

$$Ab = b_1A_{\cdot,1} +\cdots+ b_nA_{\cdot,n}.$$

Meaning: Multiplying by a vector forms a linear combination of the matrix columns.

7) Transpose

Transpose

For $A\in\mathbb{F}^{m,n}$, the transpose $A^t\in\mathbb{F}^{n,m}$ is defined by

$$(A^t)_{k,j}=A_{j,k}.$$

Meaning: Transposing a matrix exchanges its rows and columns.

8) Rank

Column Rank

The column rank of $A$ is the dimension of the span of its columns.

Meaning: Column rank measures the number of independent column directions.

Row Rank

The row rank of $A$ is the dimension of the span of its rows.

Meaning: Row rank measures the number of independent row directions.

Row Rank Equals Column Rank

For every matrix,

$$\text{row rank} = \text{column rank}.$$

Meaning: Rows and columns contain the same amount of independent linear information.

Rank

The common value is called the rank of $A$.

$$\operatorname{rank}A = \dim\operatorname{span}(\text{columns of }A).$$

Meaning: Rank measures the number of independent directions represented by a matrix.

3D. Invertibility and Isomorphisms

1) Invertible Linear Map

Invertible Linear Map

A linear map

$$T\in\mathcal{L}(V,W)$$

is invertible if there exists

$$S\in\mathcal{L}(W,V)$$

such that

$$ST=I$$

and

$$TS=I.$$

Meaning: An invertible linear map can be completely undone.

2) Inverse

Inverse

The inverse of an invertible linear map $T$ is denoted by

$$T^{-1}.$$

It satisfies

$$T^{-1}T=I$$

and

$$TT^{-1}=I.$$

Meaning: $T^{-1}$ reverses the action of $T$.

Uniqueness of the Inverse

An invertible linear map has exactly one inverse.

Meaning: There is only one linear map that perfectly reverses $T$.

3) Invertibility Criterion

Invertibility Criterion

A linear map is invertible if and only if it is both injective and surjective.

$$T\text{ invertible} \iff T\text{ injective and surjective}.$$

Meaning: A linear map is reversible exactly when no information is lost and every target vector is reached.

4) Equal-Dimension Theorem

Injectivity-Surjectivity Equivalence

If $V$ and $W$ are finite-dimensional and

$$\dim V=\dim W,$$

then

$$T\text{ invertible} \iff T\text{ injective} \iff T\text{ surjective}.$$

Meaning: Between finite-dimensional spaces of equal dimension, proving either injectivity or surjectivity is enough to prove invertibility.

5) Isomorphism

Isomorphism

An isomorphism is an invertible linear map.

Meaning: An isomorphism preserves all vector-space structure while relabeling the vectors.

Isomorphic Vector Spaces

Two vector spaces are isomorphic if there exists an isomorphism between them.

Meaning: Isomorphic vector spaces have the same linear structure.

6) Dimension and Isomorphism

Dimension Criterion for Isomorphism

Two finite-dimensional vector spaces over $\mathbb{F}$ are isomorphic if and only if

$$\dim V=\dim W.$$

Meaning: For finite-dimensional vector spaces over the same field, dimension completely determines the vector-space structure.

7) Dimension of $\mathcal{L}(V,W)$

Linear Maps and Matrices

After bases are chosen,

$$\mathcal{L}(V,W)$$

and

$$\mathbb{F}^{m,n}$$

are isomorphic, where

$$n=\dim V, \qquad m=\dim W.$$

Meaning: A linear map can be represented completely by a matrix once bases are fixed.

Dimension of the Space of Linear Maps

$$\dim\mathcal{L}(V,W) = (\dim V)(\dim W).$$

Meaning: A linear map from an $n$-dimensional space to an $m$-dimensional space requires $mn$ independent coefficients.

8) Matrix of a Vector

Matrix of a Vector

If $v_1,\ldots,v_n$ is a basis of $V$ and

$$v=b_1v_1+\cdots+b_nv_n,$$

then

$$\mathcal{M}(v) = \begin{pmatrix} b_1\\ \vdots\\ b_n \end{pmatrix}.$$

Meaning: The matrix of a vector is its coordinate column relative to the chosen basis.

9) Linear Maps as Matrix Multiplication

Matrix Representation of a Linear Map

For $T\in\mathcal{L}(V,W)$,

$$\mathcal{M}(Tv) = \mathcal{M}(T)\mathcal{M}(v).$$

Meaning: Once bases are chosen, applying a linear map is the same as multiplying by a matrix.

10) Rank of a Linear Map

Range and Matrix Rank

$$\dim\operatorname{range}T = \operatorname{rank}\mathcal{M}(T).$$

Meaning: The rank of the matrix equals the dimension of the output space actually reached by the linear map.

11) Identity and Inverse Matrices

Identity Matrix

The identity matrix has $1$ on its diagonal and $0$ elsewhere.

$$I= \begin{pmatrix} 1 & & 0\\ & \ddots &\\ 0 & & 1 \end{pmatrix}.$$

Meaning: Multiplying by the identity matrix leaves coordinates unchanged.

Invertible Matrix

A square matrix $A$ is invertible if there exists a matrix $A^{-1}$ such that

$$AA^{-1}=A^{-1}A=I.$$

Meaning: An invertible matrix represents a reversible linear transformation.

12) Change of Basis

Change-of-Basis Formula

Matrices representing the same linear operator in two different bases are related by

$$A=C^{-1}BC.$$

Meaning: Changing a basis changes the matrix representation but not the underlying linear operator.

3E. Products and Quotients of Vector Spaces

1) Product of Vector Spaces

Product of Vector Spaces

For vector spaces $V_1,\ldots,V_m$,

$$V_1\times\cdots\times V_m = \{(v_1,\ldots,v_m)\mid v_k\in V_k\}.$$

Addition and scalar multiplication are performed componentwise.

Meaning: A product space combines vectors from several vector spaces into one ordered tuple.

2) Dimension of a Product

Product Dimension

If $V_1,\ldots,V_m$ are finite-dimensional, then

$$\dim(V_1\times\cdots\times V_m) = \dim V_1+\cdots+\dim V_m.$$

Meaning: Independent directions from the component spaces simply add together.

3) Product and Direct Sum

Direct Sum Criterion

Define

$$\Gamma: V_1\times\cdots\times V_m \to V_1+\cdots+V_m$$

by

$$\Gamma(v_1,\ldots,v_m) = v_1+\cdots+v_m.$$

Then

$$V_1+\cdots+V_m$$

is a direct sum if and only if $\Gamma$ is injective.

Meaning: A sum is direct exactly when each vector has only one decomposition into components.

Dimension Criterion for a Direct Sum

For finite-dimensional subspaces,

$$V_1+\cdots+V_m$$

is a direct sum if and only if

$$\dim(V_1+\cdots+V_m) = \dim V_1+\cdots+\dim V_m.$$

Meaning: Dimensions add exactly when the subspaces contain no redundant directions.

4) Translate

Translate

For $v\in V$ and $U\subseteq V$,

$$v+U = \{v+u\mid u\in U\}.$$

Meaning: A translate shifts every vector in $U$ by the same vector $v$.

5) Quotient Space

Quotient Space

If $U$ is a subspace of $V$, then

$$V/U = \{v+U\mid v\in V\}.$$

Meaning: A quotient space treats vectors that differ by an element of $U$ as equivalent.

Equality of Cosets

For $v,w\in V$,

$$v+U=w+U \iff v-w\in U.$$

Meaning: Two vectors represent the same element of $V/U$ exactly when their difference lies in $U$.

6) Operations on a Quotient Space

Addition

$$(v+U)+(w+U) = (v+w)+U.$$

Meaning: Cosets are added using representatives.

Scalar Multiplication

$$\lambda(v+U) = (\lambda v)+U.$$

Meaning: Scalar multiplication is performed on a representative of the coset.

Quotient Space Is a Vector Space

With these operations,

$$V/U$$

is a vector space.

Meaning: Cosets of a subspace behave like vectors under the induced operations.

7) Quotient Map

Quotient Map

The quotient map

$$\pi:V\to V/U$$

is defined by

$$\pi(v)=v+U.$$

Meaning: The quotient map sends each vector to its equivalence class modulo $U$.

8) Dimension of a Quotient Space

Quotient Dimension Formula

If $V$ is finite-dimensional,

$$\dim(V/U) = \dim V-\dim U.$$

Meaning: Taking the quotient removes the directions contained in $U$.

9) Quotient by a Null Space

Induced Map

For $T\in\mathcal{L}(V,W)$, define

$$\widetilde{T}: V/\operatorname{null}T \to W$$

by

$$\widetilde{T} (v+\operatorname{null}T) = T(v).$$

Meaning: Quotienting by the null space removes exactly the information that $T$ loses.

Fundamental Isomorphism

$$V/\operatorname{null}T \cong \operatorname{range}T.$$

Meaning: After collapsing the null space, the remaining domain has exactly the same linear structure as the range.

3F. Duality

1) Linear Functional

Linear Functional

A linear functional on $V$ is a linear map

$$\varphi:V\to\mathbb{F}.$$

Meaning: A linear functional converts a vector into a scalar while preserving linear structure.

2) Dual Space

Dual Space

The dual space of $V$, denoted by $V'$, is

$$V' = \mathcal{L}(V,\mathbb{F}).$$

Meaning: The dual space contains all linear functionals on $V$.

Dimension of the Dual Space

If $V$ is finite-dimensional, then

$$\dim V'=\dim V.$$

Meaning: A finite-dimensional vector space and its dual have the same number of independent directions.

3) Dual Basis

Dual Basis

Suppose

$$v_1,\ldots,v_n$$

is a basis of $V$.

The dual basis

$$\varphi_1,\ldots,\varphi_n$$

is defined by

$$\varphi_j(v_k) = \begin{cases} 1, & j=k,\\ 0, & j\neq k. \end{cases}$$

Meaning: Each dual basis functional extracts one coordinate associated with the original basis.

Dual Basis Gives Coordinates

If

$$v = c_1v_1+\cdots+c_nv_n,$$

then

$$c_j=\varphi_j(v).$$

Meaning: Applying $\varphi_j$ to a vector returns its $j$th coordinate.

Dual Basis Is a Basis

The dual basis of a basis of $V$ is a basis of $V'$.

Meaning: The coordinate-extracting functionals span the entire dual space without redundancy.

4) Dual Map

Dual Map

For

$$T\in\mathcal{L}(V,W),$$

the dual map

$$T'\in\mathcal{L}(W',V')$$

is defined by

$$T'(\varphi) = \varphi\circ T.$$

Meaning: A dual map transforms functionals by composing them with the original linear map.

Direction Reversal

If

$$T:V\to W,$$

then

$$T':W'\to V'.$$

Meaning: Taking the dual reverses the direction of a linear map.

5) Algebra of Dual Maps

Addition

$$(S+T)'=S'+T'.$$

Meaning: Taking the dual preserves addition.

Scalar Multiplication

$$(\lambda T)'=\lambda T'.$$

Meaning: Taking the dual preserves scalar multiplication.

Composition

$$(ST)'=T'S'.$$

Meaning: Taking the dual reverses the order of composition.

6) Annihilator

Annihilator

For a subspace $U$ of $V$, the annihilator of $U$ is

$$U^0 = \{\varphi\in V' \mid \varphi(u)=0 \text{ for every }u\in U\}.$$

Meaning: The annihilator contains all linear functionals that vanish on $U$.

Annihilator Is a Subspace

$$U^0$$

is a subspace of $V'$.

Meaning: Functionals that vanish on $U$ form their own vector space.

7) Dimension of an Annihilator

Annihilator Dimension

If $V$ is finite-dimensional, then

$$\dim U^0 = \dim V-\dim U.$$

Meaning: The larger $U$ is, the fewer independent functionals can vanish on it.

Extreme Cases

$$U^0=\{0\} \iff U=V$$

and

$$U^0=V' \iff U=\{0\}.$$

Meaning: Only the zero functional vanishes on all of $V$, while every functional vanishes on the zero subspace.

8) Null Space and Range of the Dual Map

Null Space of the Dual Map

For finite-dimensional $V$ and $W$,

$$\operatorname{null}T' = (\operatorname{range}T)^0.$$

Meaning: A functional is killed by $T'$ exactly when it vanishes on the range of $T$.

Range of the Dual Map

$$\operatorname{range}T' = (\operatorname{null}T)^0.$$

Meaning: The range of the dual consists exactly of functionals that vanish on the null space of $T$.

Rank Equality

$$\dim\operatorname{range}T' = \dim\operatorname{range}T.$$

Meaning: A linear map and its dual have the same rank.

9) Injectivity and Surjectivity of Dual Maps

Surjectivity-Dual Injectivity

$$T\text{ surjective} \iff T'\text{ injective}.$$

Meaning: Reaching every vector in the target corresponds to the dual map losing no functional information.

Injectivity-Dual Surjectivity

$$T\text{ injective} \iff T'\text{ surjective}.$$

Meaning: Losing no vector information corresponds to the dual map reaching every functional.

10) Matrix of a Dual Map

Transpose Theorem

With corresponding bases and dual bases,

$$\mathcal{M}(T') = \mathcal{M}(T)^t.$$

Meaning: The matrix representation of the dual map is the transpose of the matrix representation of the original map.