A linear transformation (or linear map) is a function T:V →W, where V and W are vector spaces over the same field F, such that for all u,v∈V and all scalars c∈F:
1.T(u+v)=T(u)+T(v)(Additivity)
2.T(cu)=cT(u)(Homogeneity)
These two properties ensure that linear transformations preserve the linear structure of vector spaces.

1. Identity Transformation:
   T(x)=x,∀x∈Rn
2. Zero Transformation:
   T(x)=0,∀x∈Rn
3. Scaling Transformation:
   T(x)=λx,λ∈R
4. Rotation in R2:
   T=egin{bmatrix}cosθ & -sinθ\sinθ & cosθ\end{bmatrix}egin{bmatrix}x\y\end{bmatrix}
5. Projection onto a Line or Plane.

If T:Rn→Rm is a linear transformation, then there exists a unique matrix A∈Rm×n such that:
T(x)=Ax,∀x∈Rn
This matrix is called the standard matrix of the linear transformation. If the basis of the domain and codomain is standard, then:
A=[T(e1)T(e2)…T(en)] where ei are the standard basis vectors of R^n.

Kernel (Null Space): The kernel of T, denoted by ker(T), is the set of all vectors in V that are mapped to the zero vector in W:
ker(T)={v∈V ∣T(v)=0}
It is a subspace of the domain V.
Image (Range): The image or range of T, denoted by Im(T), is the set of all vectors in W that are images of vectors in V:
Im(T)={T(v)∣v∈V }
It is a subspace of the codomain W.

For a linear transformation T:Rn→Rm, the rank of T is the dimension of its image, and the nullity is the dimension of its kernel.
The Rank-Nullity Theorem states:
dim(kerT)+dim(ImT)=dim(V)
Or in terms of matrices:
nullity(A)+rank(A)=n
This theorem is crucial for analyzing the solvability and behavior of linear systems.

If T1:U→V and T2:V→W are linear transformations, then their composition T2∘T1:U→W is defined by:
(T2∘T1)(u)=T2(T1(u))
Properties:
- The composition of two linear transformations is also a linear transformation.
- If T1 and T2 have matrix representations A and B, respectively, then:
[T2∘T1]=BA

A linear transformation T:V →W is invertible if there exists another linear transformation S:W→V such that:
S∘T=I , T∘S=I
In terms of matrices:
If A∈Rn×n is the matrix of T, then T is invertible iff det(A)≠0, and the inverse transformation is represented by A−1.

Linear transformations in R2 and R3 can be visualized as operations such as:
- Rotation
- Reflection
- Scaling
- Shearing
- Projection
These transformations can change the orientation, length, or position of vectors while preserving linearity. Civil engineers often encounter such operations in structural modeling, mechanics, and computer simulations.

A system of linear equations can be viewed as a linear transformation: Ax=b ⇒ T(x)=b
- The solution exists iff b ∈ Im(T)
- The solution is unique iff ker(T)={0}
This perspective is fundamental in understanding the solvability and structure of linear systems in applied engineering contexts.

• Structural Analysis: Displacement and force transformations in trusses and frames.
• Finite Element Method (FEM): Transformation of stiffness matrices.
• Coordinate Transformations: Switching between local and global coordinate systems.
• CAD Modelling: Rotations, scaling, and projections used in designing components.
• Stress-Strain Relations: Linear mappings between stress and strain tensors in elasticity theory.

Linear transformations can be represented differently depending on the basis used for the vector space. This concept is crucial in simplifying problems or interpreting data from different reference frames.
Change of Basis: Let T:V →V be a linear transformation and suppose B={v1,…,vn} and B′={v′1,…,v′n} are two different bases for V. Let P be the change of basis matrix from B to B′. Then:
This transformation of matrix representations under different bases is called similarity.
Similarity of Matrices: Two matrices A and B are similar if there exists an invertible matrix P such that:
B=P−1AP
Similar matrices represent the same linear transformation under different bases. They have:
- The same determinant
- The same trace
- The same characteristic polynomial and eigenvalues.

An important class of linear transformations are those that scale vectors instead of changing their direction.
Given a linear transformation T:V →V, a non-zero vector v∈V is called an eigenvector of T if:
T(v)=λv for some scalar λ∈F, which is called the eigenvalue corresponding to v.
Finding Eigenvalues and Eigenvectors: Let A be the matrix of the linear transformation T. The eigenvalues satisfy:
det(A−λI)=0
This is called the characteristic equation. Solving it gives the eigenvalues λ1,λ2,…,λn. For each i, the eigenvectors are found by solving:
(A−λiI)x=0

A square matrix A∈Rn×n is diagonalizable if there exists an invertible matrix P and a diagonal matrix D such that:
A=PDP−1
This is equivalent to saying that the linear transformation has n linearly independent eigenvectors.
Conditions for Diagonalizability:
- Matrix A has n distinct eigenvalues ⇒ always diagonalizable.
- If not all eigenvalues are distinct, check for linearly independent eigenvectors.
Geometrical Meaning: Diagonalization simplifies the transformation into scalings along specific directions (eigenvectors). For example, in a vibrating beam, diagonalization simplifies coupled motion equations into independent modes.

A linear operator is a linear transformation T:V →V on a single vector space.
Matrix powers of linear operators are useful in recurrence relations, system modeling, and iterative methods.
Matrix Powers: If T(x)=Ax, then repeatedly applying T gives:
Tk(x)=Akx
This is used in:
- Dynamic systems: Modeling population growth, material degradation, etc.
- Iterative Solvers: Successive approximations using power methods.

In many physical systems, especially in civil engineering (e.g., vibrations of a bridge, thermal conduction in a beam), systems of differential equations arise, which can be written using linear transformations.
System of ODEs:
\[ \frac{dx}{dt} = Ax \]
Here, A is the matrix representing a linear transformation. The solution involves:
\[ x(t)=e^{At}x(0) \]
Where e^{At} is the matrix exponential, which may be computed via diagonalization or Jordan forms.
Practical Examples:
- Heat conduction modeled using Fourier’s law (linear diffusion operator)
- Frame deflection using beam bending equations (linear elasticity)
- Modal vibration analysis (linear system with eigen-decomposition)

In Finite Element Analysis (FEA), coordinate transformations are used extensively:
Local to Global Coordinate Transformations
To assemble the global stiffness matrix, each local element matrix must be transformed:
K(global)=TTK(local)T
Where T is the transformation matrix depending on element orientation.
Affine Transformations
Used to map:
- Reference elements (e.g., unit triangles) to physical elements in meshes.
- Jacobian matrices define these mappings, and their determinants indicate area or volume scaling.

A transformation T is orthogonal if its matrix A satisfies:
AT A=I⇒A−1 =AT
Orthogonal transformations preserve:
- Lengths: ∥T(x)∥=∥x∥
- Angles: ⟨T(x),T(y)⟩=⟨x, y⟩
Examples:
- Rotations (no distortion, used in simulations)
- Reflections (used in symmetry analysis)
Relevance to Civil Engineering:
- Used in aligning axes in structural design
- Important in computer graphics for CAD software
- Ensure numerical stability in simulations (QR decomposition)