What's a real world application of linear algebra?

Almost all of us, have used image resizing.

When you resize an image on your phone, you're actually using linear algebra!

The scaling operation can be represented as a simple matrix multiplication:

[1.5  0  ] [x] = [1.5x]
[ 0   1.5] [y]   [1.5y]

What is Linear Algebra?

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Algebra:

Branch of mathematics that uses letters & symbols to represent numbers & quantities in an equations. It's about finding unknown values & understanding relationships between variables.

Example:

2x + 5 = 13

In this algebraic equations:

1. x is an unknown value we want to find
2. 2, 5, 13 are known numbers
3. the goal is to solve for x

solving this equations:

1. subtract 5 on both sides
2. 2x = 8
3. divided by 2 on both sides
4. x = 4

Breaking down what algebra is:

1. Use of symbols 'x' to represent unknown numbers
2. A relationship between known & unknown quantities.

Algebra allows us to:

1. Represent general rules & relationship mathematically.
2. Solve problems by manipulating equations
3. Understand patterns and make predictions.

What is linear?

In mathematics, linear refers to a relationship or function that forms a straight line when graphed. It involves quantities that are directly proportional to each other. The key properties of linearity are:

1. Additivity: f(x+y) = f(x) + f(y)
2. Homogeneity: f(ax) = a f(x), where 'a' is a constant.

Now let's introduce more concept on linear algebra:

1. Vectors
2. Matrices

Vectors and matrices are mathematical concepts that exist independently of linear algebra. They are used in various areas of mathematics & science.

However, linear algebra as a field heavily utilises & develops these concepts. It's not linear algebra borrowed these tools, but rather it grew around them & with them.

Linear Algebra:

Heavily utilises and develops concepts of vectors and matrices

Linear algebra provides a framework that:

1. Formalises how we use vectors & matrices
2. Provides precise definition for vectors, matrices & related concepts.
3. Define vectors not just as an arrow, but as an element of a vector space with specific properties.
4. Rules of operations:
5. Establishes clear rules for how to perform operations with vector & matrices.
6. How to add vectors?
7. How to multiply vectors?
8. What conditions must be met for these operations to be valid?
9. Explore their properties in depth
10. Vector properties like identifying minimal sets of vectors that can generate an entire vector space
11. Matrix properties like
12. Determinants: A special number calculated from a square matrix, revealing important information about the matrix.
13. Eigenvalues & eigenvectors: Special scalars & vectors associated with a matrix
14. Transformation: how matrices represent linear transformations between vector spaces.
15. Develops new ways to apply them to solve problems.

Vectors and Matrices in Linear Algebra

1. Vectors are studied more abstractly, leading to the concept of vector spaces.
2. Matrices are used to represent linear transformations & solve systems of equations.

While vectors & matrices existed before linear algebra was formalised as a field, linear algebra has significantly expanded our understanding & use of these tools.

Solving Force Vectors: Wind and Gravity Example

Vector:

y
        ^
     4  |    F1 (Wind)
        |     \
     3  |      \  Resultant Force
        |       \ |
     2  |        \|
        |         •-----> x
     1  |        /|
        |       /
     0  |      /
        |     /
    -1  |    /
        |   /
    -2  |  /
        | /  F2 (Gravity)
        |/
    -3  +-------------------

Forces:
F1 (Wind):      3 units right, 2 units up
F2 (Gravity):   4 units down, 1 unit right

Object: •

1. F1 represents a wind force pushing the object up and to the right.
2. F2 represents the force of gravity pulling the object down and slightly to the right (e.g., if on a slight incline).
3. The resultant force shows the overall effect of both forces combined.

Given:

• F1 (Wind): 3 units right, 2 units up
• F2 (Gravity): 4 units down, 1 unit right

Step 1: Express vectors in (x, y) format

• F1 = (3, 2)
• F2 = (1, -4)

Step 2: Add vectors to find the resultant

Resultant

= F1 + F2 
= (3, 2) + (1, -4) 
= (3 + 1, 2 + (-4)) 
= (4, -2)

Step 3: Interpret the result

The resultant force is 4 units to the right and 2 units down.

Step 4: Calculate the magnitude of the resultant force

Magnitude 
= √(x² + y²) 
= √(4² + (-2)²) 
= √(16 + 4) 
= √20 
≈ 4.47 units

Step 5: Calculate the angle of the resultant force

Angle 
= tan⁻¹(y/x) 
= tan⁻¹(-2/4) 
≈ -26.57°

The negative angle indicates it's below the horizontal.

Conclusion:

The object will move with a force of approximately 4.47 units at an angle of 26.57° below the horizontal to the right.

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