Page 1 of 18

Vectors:

• Definition: A magnitude showing steps in some direction.

  $$\mathbf{u} = \begin{bmatrix} 1 \\ -1 \end{bmatrix}$$

Operations:

• Addition:

  $$\mathbf{v_1} = \begin{bmatrix} 2 \\ 3 \end{bmatrix}, \quad \mathbf{v_2} = \begin{bmatrix} 1 \\ 1 \end{bmatrix}$$

---

Special Vectors:

• A vector has some special vectors called basis vectors. Example:

  $$\mathbf{v} = 2\hat{i} + 3\hat{j}$$

  (Unit vector in $\hat{i}$ and directions, respectively).

---

Page 2 of 18

Matrices:

Linear Transformation:

• A matrix defines where a vector lands after transformation.
  Example: $$\text{Input Vector: } \begin{bmatrix} x \\ y \end{bmatrix} \rightarrow \begin{bmatrix} 3 & 2 \\ -2 & 4 \end{bmatrix} \begin{bmatrix} x \\ y \end{bmatrix} = \begin{bmatrix} 3x + 2y \\ -2x + 4y \end{bmatrix}$$

Matrix Multiplication:

• Composition of transformations (e.g., shear $\rightarrow$ rotation $\rightarrow$ final linear transformation):

  $$\begin{bmatrix} n & y \\ k & m \end{bmatrix} \cdot \begin{bmatrix} z & w \\ q & r \end{bmatrix}$$

---

Page 3 of 18

3D Transformation:

• Linear Transformation in 3D: $$\begin{bmatrix} x \\ y \\ z \end{bmatrix} \rightarrow \begin{bmatrix} x' \\ y' \\ z' \end{bmatrix}$$

Determinant of Transformation:

• How much the area (or volume) is scaled after transformation: $$\text{Initial Matrix: } \begin{bmatrix} 2 & 0 \\ 1 & 5 \end{bmatrix}, \quad |A| = 2$$

---

Page 4 of 18

Applications of Matrices:

Linear System of Equations:

• Example: $$2x + 5y + 3z = -3 \\ 4x + 0y + 8z = 0 \\ 1x + 3y + 0z = 2$$

Solving:

• Representing as a matrix: $$\begin{bmatrix} 2 & 5 & 3 \\ 4 & 0 & 8 \\ 1 & 3 & 0 \end{bmatrix} \begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} -3 \\ 0 \\ 2 \end{bmatrix}$$

Cases:

1. Case $|A| \neq 0$:
   Only one unique solution exists:

   $$A\mathbf{x} = \mathbf{v} \quad \Rightarrow \quad \mathbf{x} = A^{-1} \mathbf{v}$$

---

Page 5 of 18

Important Conclusion:

( A^{-1} ) is defined only when ( |A| \neq 0 ).

---

Non-Square Matrices:

Transformation:

For a non-square matrix ( 2 \times N ), the transformation is as follows: [ \begin{bmatrix} 2 \times N \text{ input} \end{bmatrix} \rightarrow L(V) \rightarrow \begin{bmatrix} 3 \times 1 \text{ output} \end{bmatrix} ]

Example:

[ \begin{bmatrix} 2 & 0 \ 1 & 1 \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix}

\begin{bmatrix} 2x \ x + y \end{bmatrix} ]

---

Dot Products:

Formula:

[ \vec{u} \cdot \vec{v} = nv ]

Diagram:

graph TD
    A[Vector u] --> B[Vector v]
    A --> C[Vector n]

Explanation:

• If the vectors point in the same direction, their dot product is positive.
• Else, if they point in different directions, the dot product is negative.
• If ( \theta = 90^\circ ), then ( \vec{u} \cdot \vec{v} = 0 ).

---

Page 6 of 18

Cross Products:

Formula:

[ \vec{u} \times \vec{\omega} = \begin{bmatrix} i & j & k \ a & b & c \ x & y & z \end{bmatrix} = \begin{bmatrix} i(x - z) - j(y - z) + k(-x + b) \end{bmatrix} ]

Magnitude:

[ |\vec{u} \times \vec{\omega}| = |\vec{u}| |\vec{\omega}| \sin \theta ]

Direction:

The direction is lost to the plane represented by ( \vec{u} \times \vec{\omega} ).

---

Page 7 of 18

Cramer's Rule:

Diagram:

graph TD
    A[Area of triangle] --> B[Area scaled by det(A)]
    B --> C[New Area]

Explanation:

Remember, all areas are scaled by some ( n )-value that is given by ( |A| ).

Formula:

[ \text{New Area} = | \text{det}(A) | \times \text{Old Area} ]

Example:

Given equations: [ 2x - y = 4 \quad \text{and} \quad x + y = 2 ]

Matrix Representation: [ \begin{bmatrix} x & y \end{bmatrix}

\begin{bmatrix} 2 & 4 \ 0 & 2 \end{bmatrix} ]

Finding Determinants: [ \text{Area}{xy} = \text{det} \begin{bmatrix} 0 & 1 \end{bmatrix} \quad \text{and} \quad \text{Area}{xy} = \text{det} \begin{bmatrix} 2 & -1 \end{bmatrix} ]

---

Page 8 of 18

Translation of Coordinate System:

Diagram:

graph TD
    A[Coordinate a, b] --> B[Coordinate o_x, o_y]
    B --> C[Translated System]

Explanation:

Why different coordinate systems?

• Axes are just a construct and can have different perspectives.

Example:

Translate ( 90^\circ ) rotation in her language:

1. Step 1: Go to our language.
2. Step 2: Rotate ( 90^\circ ).
3. Step 3: Go back to her language.

Matrix Representation: [ M' = M \begin{bmatrix} Jx & Jy \end{bmatrix} \rightarrow \begin{bmatrix} o_x' & o_y' \end{bmatrix} \quad \text{90° rotation in her language.} ]

---

Page 9 of 18

Eigen-Vectors & Eigen-Values

An eigen vector is a non-zero vector that changes at most by a scalar factor when a linear transformation is applied to it.
The scalar factor is called the eigen value.

Transformation Diagrams:

graph LR
    A[Original Vector] --> B[Transformation]
    C[Transformed Vector] --> D[This vector remained on its own span]

Example:

A 3D rotation (a linear transformation):
An eigen vector represents the axis of rotation.

Scalar Multiplication:

Let:

A * v = λ * v

Where:

• ( v ): Some eigen vector
• ( \lambda ): Eigen value

To find ( \lambda ) and ( v ):
$A \vec{v} = (\lambda I) \vec{v}$ True for zero vector.

For non-zero vectors:

(INSERT IMAGE)

---

Page 10 of 18

Eigen Values:

Let: $B \vec{v} = 0$
Where ( B = A - \lambda I ).

We need to find transformations that squish space into lower dimensions.
That is: $|B| = 0$
or:

$$\begin{bmatrix} x_1 - \lambda x_2 & x_1 x_2 \\ y_1 - \lambda y_2 & y_1 y_2 \end{bmatrix}$$

$= 0$

Key Notes:

• Find det ( \lambda ) (eigen value).
• Number of eigen values: ( \leq 2 ).
• Number of eigen vectors: ( [0, \infty) ).

---

Vector Spaces:

Functions are also like vectors.
They can be added or scaled; the only difference is that coordinates are infinitely large.

Formal Definition of Linearity:

A linear operator/transformation being linear:

1. Additivity:
   $L(\vec{v} + \vec{w}) = L(\vec{v}) + L(\vec{w})$

Example: Derivative is a linear operator.

---

Page 11 of 18

Functions as Matrices:

(INSERT IMAGE)

Matrix representation:

[ a_n ]
[ a_{n-1} ]
[ a_{n-2} ]
[ ... ]
[ a_0 ]

Polynomial Representation:

$$a_n x^n + a_{n-1} x^{n-1} + ... + a_0$$

---

Page 12 of 18

Essence of Calculus

Introduction:

Diagram Representation:

graph TD
    A[Area of Circle] --> B[Thickness]
    C[Rotated Region] --> D[Approximation]

Area of Circle:

$\int \text{revolves}$
Assume it's rotated with 2nd axis as dimensions: $\pi r^2$

If ( dx \to 0 ), this approximation seems off.

---

Breaking Down Problems:

Any problem can be broken down into the sum of many small values.

Finding Area Between Curves:

Rectangle: $\text{Area} = \frac{1}{2} b h$

For other curves, calculus provides tools:

• Every curve for ( dn \to 0 ) gives a rectangle: $dA = x^2 \cdot dn$

An inverse feature between differentiation and integration: $\frac{dA}{dn} = x^2$

---

Page 13 of 18

Derivative

Definition: Best constant approximation of rate of change around a point.


$\frac{ds(t)}{dt} \approx \frac{s(t + dt) - s(t)}{dt}$

• Velocity is the tiny change in distance over time, caused by the change.

Pure Math

As ( dt \to 0 ), this gives a tangent to the graph at a single point. Without velocity, the change is gone.

$$\frac{ds}{dt} = \frac{3t + dt^3}{dt} \quad \text{(Ignore } dt^3 \text{ as } dt \to 0 \text{)}$$ $$\frac{ds}{dt} = 3t^2$$

---

Real-World Applications

• Most real-world applications revolve around polynomials, oscillations, or pure functions.
• This gives us a reason to find derivatives of common problems.

Example:

$\frac{d(\sqrt{t})}{dt}$

---

Page 14 of 18

Example 2:

Derivative of Square Root
Given ( \frac{d\sqrt{x}}{dx} ):

$$\frac{x + dx - dx}{\sqrt{(x + dx)\sqrt{x}}}$$

As ( dx \to 0 ):

$$\frac{\sqrt{dx} + \sqrt{x}}{2\sqrt{x}}$$

Therefore: $\frac{dy}{dx} = \frac{-1}{x^{3/2}}$

---

Trigonometric Derivatives

Derivative of ( \sin(\theta) ):

$\frac{d(\sin\theta)}{d\theta} = \cos(\theta)$

Derivative of ( \cos(\theta) ):

$\frac{d(\cos\theta)}{d\theta} = -\sin(\theta)$

---

Page 15 of 18

Chain Rule & Product Rule

Chain Rule:

$\frac{d(f(g))}{dn} = \frac{df}{dg} \cdot \frac{dg}{dn}$

For ( y = \sin(x + n) ): $\frac{dy}{dx} = \cos(x + n) \cdot \frac{dg}{dn}$

---

Product Rule:

$\frac{d(fg)}{dn} = f \cdot \frac{dg}{dn} + g \cdot \frac{df}{dn}$

Example:

$$\frac{d(\sin x \cdot \cos x)}{dx} = \sin x \cdot \frac{d(\cos x)}{dx} + \cos x \cdot \frac{d(\sin x)}{dx}$$

Composition Rule:

$$\frac{d(f(g))}{dn} = \frac{d}{dx} \cdot \frac{dg}{dn}$$

---

Euler's Number (( e ))

Derivative of Exponentials:

$\frac{d(2^t)}{dt} = 2^t \cdot \ln(2)$

---

Page 16 of 18

Logarithmic Derivatives

Derivative of ( e^t ):

$\frac{d(e^t)}{dt} = e^t$

For ( \log_e 2 ): $\frac{d(e^{t})}{dt} = 1$

---

Implicit Differentiation

For ( y = \ln x ):

$$x = e^y \quad \Rightarrow \quad \frac{dx}{dy} = e^y$$

And:

$$\frac{dy}{dx} = \frac{1}{e^y} = \frac{1}{x}$$

---

Limits: ( L'Hôpital's Rule ) & Epsilon-Delta Definition

Formal Definition of Derivative:

$$\frac{df}{dx} = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h}$$

Formal Definition of Limits:

For a real-valued function ( f ) on a limit point ( c ), we say: $\lim_{x \to c} f(x) = L$

If for every ( \epsilon > 0 ), there exists ( \delta > 0 ) so that ( |x - c| < \delta ) implies ( |f(x) - L| < \epsilon ).

( L'Hôpital's Rule ):

$$\lim_{x \to a} \frac{f'(x)}{g'(x)} = \frac{f(a)}{g(a)}$$

---

Page 17 of 18

Integral


The graph depicts the integration of the function ( v(t) = (3 - t)t ).

Definition:

$$\int v(t) \, dt \quad (\text{Sum of areas})$$

• Significance: Sum approaching as ( \Delta t \to 0 ).

General Form:

$$\int_a^b f(x) \, dx = F(b) - F(a)$$

• Derivative relationship:

$$\frac{dF(x)}{dx} = f(x)$$

---

Application of Integral:

Average of a Continuous Function:

To find the average of ( \sin(x) ):

$$\text{Avg. } \sin(x) = \frac{1}{\pi} \int_0^\pi \sin(x) \, dx$$

Solving:

$$= \frac{1}{\pi} \left[ -\cos(x) \right]_0^\pi$$ $$= \frac{1 + 1}{\pi} = \frac{2}{\pi}$$

---

Page 18 of 18

Taylor Series

Definition:

Study to find polynomial approximations of non-polynomial functions.

Example:

Let ( f(x) = \cos(x) ), and ( P(x) = c_0 + c_1x + c_2x^2 ).

At ( x = 0 ):

• ( f(0) = 1 ), so:

$$c_0 = 1$$

• ( \frac{d}{dx} f(x) \big|_{x=0} = 0 ), so:

$$c_1 = 0$$

• ( \frac{d^2}{dx^2} f(x) \big|_{x=0} = -1 ), so:

$$c_2 = \frac{-1}{2}$$

Final Polynomial:

$$P(x) = f(0) + \frac{d}{dx} f(0) \cdot x + \frac{d^2}{dx^2} f(0) \cdot \frac{x^2}{2!} + \frac{d^3}{dx^3} f(0) \cdot \frac{x^3}{3!} + \ldots$$

---

References:

• Linear Algebra Basics
• Matrix Determinants and Applications
• Coordinate System Translation Techniques
• Integration techniques