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.

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

Unit Vector — a vector whose length =1

$$u = v/||v||$$

Tips

• Do one transformation at a time from right to left