Linear Transformations: Definitions, Properties & Applications

1. Introduction to Linear Transformations

1.1 Definition

A linear transformation (or linear map) $T:V\to W$ between two vector spaces $V$ and $W$ over the same field $\mathbb{F}$ satisfies two properties for all $\mathbf{u},\mathbf{v}\in V$ and $c\in \mathbb{F}$:

1. Additivity (Preservation of Vector Addition): $T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$
2. Homogeneity (Preservation of Scalar Multiplication): $T(c\mathbf{u})=cT(\mathbf{u})$

Example: Let $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ be defined by $T(x,y)=(2x+y,x-3y)$. Check linearity:

• $T((x_1,y_1)+(x_2,y_2))=T(x_1+x_2,y_1+y_2)=(2(x_1+x_2)+(y_1+y_2),(x_1+x_2)-3(y_1+y_2))$ $=(2x_1+y_1,x_1-3y_1)+(2x_2+y_2,x_2-3y_2)=T(x_1,y_1)+T(x_2,y_2)$.
• $T(c(x,y))=T(cx,cy)=(2cx+cy,cx-3cy)=c(2x+y,x-3y)=cT(x,y)$. Thus, $T$ is linear.

1.2 Non-Examples of Linear Transformations

Transformations that do not satisfy both properties are non-linear. Examples:

• $T(x,y)=(x^2,y)$ (fails homogeneity).
• $T(x,y)=(x+1,y)$ (fails additivity).
• $T(x,y)=(e^x,y)$ (fails both).

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2. Matrix Representation of Linear Transformations

2.1 Standard Matrix of a Linear Transformation

Every linear transformation $T:{\mathbb{R}}^n\to {\mathbb{R}}^m$ can be represented by an $m\times n$ matrix $A$ such that: $$T(\mathbf{x})=A\mathbf{x}$$

Steps to Find the Standard Matrix $A$:

1. Apply $T$ to the standard basis vectors ${\mathbf{e}}_1,{\mathbf{e}}_2,\dots ,{\mathbf{e}}_n$ of ${\mathbb{R}}^n$.
2. The columns of $A$ are the images $T({\mathbf{e}}_1),T({\mathbf{e}}_2),\dots ,T({\mathbf{e}}_n)$.

Example: Let $T:{\mathbb{R}}^2\to {\mathbb{R}}^3$ be defined by: $$T(x,y)=(x+y,2x-y,x+3y)$$ Find the standard matrix $A$ of $T$:

• $T(1,0)=(1,2,1)$
• $T(0,1)=(1,-1,3)$ Thus, $$A=\left[\begin{array}{cc}1& 1\\ 2& -1\\ 1& 3\end{array}\right]$$

2.2 Verification of Linearity via Matrix Representation

If $T(\mathbf{x})=A\mathbf{x}$, then:

• $T(\mathbf{u}+\mathbf{v})=A(\mathbf{u}+\mathbf{v})=A\mathbf{u}+A\mathbf{v}=T(\mathbf{u})+T(\mathbf{v})$.
• $T(c\mathbf{u})=A(c\mathbf{u})=c(A\mathbf{u})=cT(\mathbf{u})$. Thus, all matrix transformations are linear.

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3. Properties of Linear Transformations

3.1 Kernel (Null Space) and Range (Image)

• Kernel (Null Space): $\text{ker}(T)=\{\mathbf{x}\in V\mid T(\mathbf{x})=0\}$.
  • Dimension of $\text{ker}(T)$ is called the nullity of $T$.
• Range (Image): $\text{Im}(T)=\{T(\mathbf{x})\mid \mathbf{x}\in V\}$.
  • Dimension of $\text{Im}(T)$ is called the rank of $T$.

Rank-Nullity Theorem: For a linear transformation $T:V\to W$, $$\text{rank}(T)+\text{nullity}(T)=\dim (V)$$

Example: Let $T:{\mathbb{R}}^3\to {\mathbb{R}}^2$ be defined by: $$T(x,y,z)=(x+y-z,2x-y+3z)$$ Find $\text{ker}(T)$ and $\text{Im}(T)$:

1. Solve $T(x,y,z)=(0,0)$: $$\{\begin{array}{l}x+y-z=0\\ 2x-y+3z=0\end{array}$$ Row reduce: $$\left[\begin{array}{ccc}1& 1& -1\\ 2& -1& 3\end{array}\right]\to \left[\begin{array}{ccc}1& 1& -1\\ 0& -3& 5\end{array}\right]$$ Solutions: $y=\frac{5}{3}z$, $x=-\frac{2}{3}z$. Thus, $\text{ker}(T)=\text{span}\left\{\left[\begin{array}{c}-2\\ 5\\ 3\end{array}\right]\right\}$, and $\text{nullity}(T)=1$.
2. Since $\dim ({\mathbb{R}}^3)=3$, $\text{rank}(T)=3-1=2$. The range is all of ${\mathbb{R}}^2$ because the two rows of $A$ are linearly independent.

3.2 One-to-One (Injective) and Onto (Surjective) Transformations

• Injective (One-to-One): $T$ is injective if $T(\mathbf{u})=T(\mathbf{v})$ implies $\mathbf{u}=\mathbf{v}$.
  • Equivalent to $\text{ker}(T)=\{0\}$.
• Surjective (Onto): $T$ is surjective if $\text{Im}(T)=W$.
  • Equivalent to $\text{rank}(T)=\dim (W)$.

Example: Let $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ be defined by $T(x,y)=(3x+y,x-2y)$.

• Check injectivity: Solve $T(x,y)=(0,0)$: $$\{\begin{array}{l}3x+y=0\\ x-2y=0\end{array}$$ Solution: $x=y=0$. Thus, $\text{ker}(T)=\{0\}$, so $T$ is injective.
• Check surjectivity: The matrix of $T$ is $\left[\begin{array}{cc}3& 1\\ 1& -2\end{array}\right]$, which has full rank (determinant $\ne 0$). Thus, $T$ is surjective.

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4. Applications of Linear Transformations

4.1 Computer Graphics

Linear transformations are used to:

• Rotate objects (e.g., $T(x,y)=(x\cos \theta -y\sin \theta ,x\sin \theta +y\cos \theta )$).
• Scale objects (e.g., $T(x,y)=(k_{x}x,k_{y}y)$).
• Translate objects (non-linear, but often approximated linearly in homogeneous coordinates).

Example (Rotation): Rotate a point $(1,0)$ by $90^{\circ }$ counterclockwise: $$T(x,y)=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{c}1\\ 0\end{array}\right]=\left[\begin{array}{c}0\\ 1\end{array}\right]$$

4.2 Cryptography

Linear transformations are used in linear congruential generators and matrix-based ciphers.

Example (Simple Substitution Cipher): Let $T$ map letters to numbers via a linear transformation (e.g., $A\to 0,B\to 1,\dots ,Z\to 25$) and apply a matrix $A$ modulo 26.

4.3 Data Science and Machine Learning

• Principal Component Analysis (PCA): Uses linear transformations to project data onto lower-dimensional subspaces.
• Linear Regression: Fits a linear model $y=A\mathbf{x}+\mathbf{b}$.

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5. Comparison Table: Key Concepts

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6. Common Mistakes and Clarifications

1. Assuming All Transformations Are Linear:

   • Non-linear examples: $T(x,y)=(x^2,y)$, $T(x)=\sin (x)$.
   • Fix: Always check additivity and homogeneity.

2. Confusing Kernel and Range:

   • Kernel is about inputs mapped to zero; range is about outputs covered.
   • Fix: Kernel = null space of $A$; range = column space of $A$.

3. Matrix Representation Errors:

   • Forgetting to apply $T$ to all basis vectors.
   • Fix: Use $T({\mathbf{e}}_1),T({\mathbf{e}}_2),\dots$ as columns of $A$.

4. Rank-Nullity Misapplication:

   • Applying to non-linear transformations or non-vector spaces.
   • Fix: Only valid for linear maps between finite-dimensional vector spaces.

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7. Worked Exam-Style Problems

Problem 1: Verify Linearity

Let $T:{\mathbb{R}}^2\to {\mathbb{R}}^3$ be defined by: $$T(x,y)=(x+2y,3x-y,x+y)$$ Is $T$ linear? Justify.

Solution: Check additivity and homogeneity:

1. Additivity: $$T((x_1,y_1)+(x_2,y_2))=T(x_1+x_2,y_1+y_2)=(x_1+x_2+2(y_1+y_2),3(x_1+x_2)-(y_1+y_2),(x_1+x_2)+(y_1+y_2))$$ $$=(x_1+2y_1,3x_1-y_1,x_1+y_1)+(x_2+2y_2,3x_2-y_2,x_2+y_2)=T(x_1,y_1)+T(x_2,y_2)$$
2. Homogeneity: $$T(c(x,y))=T(cx,cy)=(cx+2cy,3cx-cy,cx+cy)=c(x+2y,3x-y,x+y)=cT(x,y)$$ Thus, $T$ is linear.

Problem 2: Find the Standard Matrix

Let $T:{\mathbb{R}}^3\to {\mathbb{R}}^2$ be defined by: $$T(x,y,z)=(x-y+z,2x+y-3z)$$ Find the standard matrix $A$ of $T$.

Solution: Apply $T$ to the standard basis vectors:

• $T(1,0,0)=(1,2)$
• $T(0,1,0)=(-1,1)$
• $T(0,0,1)=(1,-3)$ Thus, $$A=\left[\begin{array}{ccc}1& -1& 1\\ 2& 1& -3\end{array}\right]$$

Problem 3: Determine Kernel and Rank

Let $T:{\mathbb{R}}^4\to {\mathbb{R}}^2$ have the standard matrix: $$A=\left[\begin{array}{cccc}1& 2& -1& 3\\ 2& 4& -2& 6\end{array}\right]$$ Find $\text{ker}(T)$ and $\text{rank}(T)$.

Solution:

1. Row reduce $A$: $$\left[\begin{array}{cccc}1& 2& -1& 3\\ 2& 4& -2& 6\end{array}\right]\to \left[\begin{array}{cccc}1& 2& -1& 3\\ 0& 0& 0& 0\end{array}\right]$$ The system $A\mathbf{x}=0$ reduces to $x_1+2x_2-x_3+3x_4=0$. Free variables: $x_2,x_3,x_4$. General solution: $$\mathbf{x}=\left[\begin{array}{c}-2x_2+x_3-3x_4\\ x_2\\ x_3\\ x_4\end{array}\right]=x_2\left[\begin{array}{c}-2\\ 1\\ 0\\ 0\end{array}\right]+x_3\left[\begin{array}{c}1\\ 0\\ 1\\ 0\end{array}\right]+x_4\left[\begin{array}{c}-3\\ 0\\ 0\\ 1\end{array}\right]$$ Thus, $\text{ker}(T)=\text{span}\left\{\left[\begin{array}{c}-2\\ 1\\ 0\\ 0\end{array}\right],\left[\begin{array}{c}1\\ 0\\ 1\\ 0\end{array}\right],\left[\begin{array}{c}-3\\ 0\\ 0\\ 1\end{array}\right]\right\}$, and $\text{nullity}(T)=3$.
2. Since $\dim ({\mathbb{R}}^4)=4$, $\text{rank}(T)=4-3=1$.

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8. Exam Tip: How to Score Full Marks

What Examiners Look For:

1. Definitions:

   • Always state the exact definition of linear transformations, kernel, range, etc.
   • Example: "A linear transformation $T:V\to W$ satisfies $T(\mathbf{u}+\mathbf{v})=T(\mathbf{u})+T(\mathbf{v})$ and $T(c\mathbf{u})=cT(\mathbf{u})$."

2. Verification of Linearity:

   • Show both additivity and homogeneity with clear algebraic steps.
   • Avoid skipping steps (e.g., expanding $T(\mathbf{u}+\mathbf{v})$).

3. Matrix Representation:

   • Always explicitly apply $T$ to the standard basis vectors.
   • Write the matrix $A$ with columns as $T({\mathbf{e}}_1),T({\mathbf{e}}_2),\dots$.

4. Kernel and Range:

   • For kernel: Solve $A\mathbf{x}=0$ and express the general solution in terms of free variables.
   • For range: State whether it’s all of ${\mathbb{R}}^m$ or a subspace, and justify using rank.

5. Applications:

   • In exam questions, connect theory to applications (e.g., "This transformation can be used in computer graphics to rotate objects").
   • Avoid vague statements; be specific (e.g., "The rotation matrix $\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]$ rotates vectors by $90^{\circ }$ counterclockwise").

Common Pitfalls to Avoid:

• Assuming linearity without verification: Always check both properties.
• Incorrect basis vectors: Use ${\mathbf{e}}_1=(1,0,\dots ,0)$, not arbitrary vectors.
• Rank-nullity misapplication: Only use it for linear transformations between finite-dimensional spaces.
• Ignoring the field: Linear transformations are defined over a specific field (usually $\mathbb{R}$ or $\mathbb{C}$).

Sample Answer Structure for Proofs:

Question: Is $T(x,y)=(x^2+y,x+y^2)$ linear?

Model Answer: To determine if $T$ is linear, we check the two properties of linearity:

1. Additivity: Let $\mathbf{u}=(x_1,y_1)$ and $\mathbf{v}=(x_2,y_2)$. $$T(\mathbf{u}+\mathbf{v})=T(x_1+x_2,y_1+y_2)=\left((x_1+x_2{)}^2+(y_1+y_2),(x_1+x_2)+(y_1+y_2{)}^2\right)$$ $$T(\mathbf{u})+T(\mathbf{v})=(x_1^2+y_1,x_1+y_1^2)+(x_2^2+y_2,x_2+y_2^2)=(x_1^2+x_2^2+y_1+y_2,x_1+x_2+y_1^2+y_2^2)$$ Clearly, $T(\mathbf{u}+\mathbf{v})\ne T(\mathbf{u})+T(\mathbf{v})$ (e.g., $(x_1+x_2{)}^2\ne x_1^2+x_2^2$). Thus, $T$ fails additivity.

2. Homogeneity: Let $c\in \mathbb{R}$. $$T(c\mathbf{u})=T(c{x}_1,c{y}_1)=(c^2{x}_1^2+c{y}_1,c{x}_1+c^2{y}_1^2)$$ $$cT(\mathbf{u})=c(x_1^2+y_1,x_1+y_1^2)=(c{x}_1^2+c{y}_1,c{x}_1+c{y}_1^2)$$ These are not equal unless $c=0$ or $c=1$, so $T$ fails homogeneity.

Conclusion: $T$ is not a linear transformation.

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9. Summary of Key Formulas

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10. Practice Questions for Revision

1. Let $T:{\mathbb{R}}^3\to {\mathbb{R}}^2$ be defined by $T(x,y,z)=(x+y-z,2x-y+3z)$.

   • Find the standard matrix of $T$.
   • Determine if $T$ is injective or surjective. Justify.

2. Verify whether the following transformation is linear: $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ defined by $T(x,y)=(x+y^2,x^2-y)$.

3. Let $A=\left[\begin{array}{cc}1& 2\\ -1& 3\end{array}\right]$ and define $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ by $T(\mathbf{x})=A\mathbf{x}$.

   • Find $\text{ker}(T)$ and $\text{Im}(T)$.
   • Is $T$ invertible? Why or why not?

4. Define a linear transformation $T:{\mathbb{R}}^2\to {\mathbb{R}}^3$ that rotates vectors in ${\mathbb{R}}^2$ by $45^{\circ }$ and maps them to ${\mathbb{R}}^3$ by adding a $z$-component of 1. Find its standard matrix.

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11. Further Reading and Resources

• Textbooks:
  • Linear Algebra and Its Applications by Gilbert Strang.
  • Introduction to Linear Algebra by Serge Lang.
• Online Resources:
  • Khan Academy’s Linear Transformations module.
  • MIT OpenCourseWare: Linear Algebra (Gilbert Strang).
• Practice:
  • Solve past TU/PU exam papers (focus on Units 2–5 for matrix-related questions).
  • Use Wolfram Alpha to verify matrix operations and transformations.

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