Linear Algebra

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.



Linear Transformation

Conditions: 1. Origin remains fixed

2. All lines remain straight

Linear Combination

  • Sum of two vectors

  • Tip-to-tail method


  • Span of 2 vectors is set of all their linear combination

Linearly dependent

Remove one point and there is no change




Product of two matrices in a single action


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.




Rank 1: Single line

Rank : 2D Plane

Rank 3: 3D plane


Natural but surprising correspondence

Cramer's rule

Gaussian Elimination

Translating between coordinate systems

  1. Translate to your own basic vectors

  2. Perform linear transformation

  3. Inverse translate basis vectors back to original


Identity matrix

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

1 0 0

0 1 0

0 0 1


Unit Vector — a vector whose length =1

u=v/vu = v/||v||


  • Do one transformation at a time from right to left

Last updated