# Linear Algebra

Last updated

Last updated

Why linear algebra

Translation and manipulation of space and data

We usually understand 3D objects such as a primitive cube as a single mass but the reality is that they are made of points. A cube has at least 4 points. Each point has it whole XYZ coordinates. How can we perform transformations across all these points. The answer is using matrices/linear transformation. A matrix, in a way, is like a rule, that can be apply to each point.

Unit Vector — a vector whose length =1

$u = v/||v||$

**Tips**

Do one transformation at a time from right to left

Item

Cheat

**Linear Transformation**

**Conditions:
**1. Origin remains fixed

2. All lines remain straight

Linear Combination

Sum of two vectors

Tip-to-tail method

Span

Span of 2 vectors is set of all their linear combination

Linearly dependent

Remove one point and there is no change

Scalars

numbers

Composition

Product of two matrices in a single action

Determinant

Scaling factor

if negative, orientation is flipped

det(m₁m₂) = det(m₁)det(m₂)

**Dot product**

if dot product of 2 vector = 0, then vectors are perpendicular

**Cross product**

Basic vectors

i-hat - unit vector of x j-hat - unit vector of y

The basis of vector spaces is set of linearly independent vectors that span the full span.

Eigenvalue

Eigenvector

Rank

Rank 1: Single line

Rank : 2D Plane

Rank 3: 3D plane

Duality

Natural but surprising correspondence

Cramer's rule

Gaussian Elimination

Translating between coordinate systems

Translate to your own basic vectors

Perform linear transformation

Inverse translate basis vectors back to original

A⁻¹MA

Identity matrix

Multiplying any matrix by the *identity matrix* results in the same matrix.

1 0 0

0 1 0

0 0 1

Kernel