# 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 fixed2. 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 flippeddet(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 yThe basis of vector spaces is set of linearly independent vectors that span the full span. ​ Eigenvalue ​ ​ Eigenvector ​ ​ Rank Rank 1: Single lineRank : 2D PlaneRank 3: 3D plane ​ Duality Natural but surprising correspondence ​ Cramer's rule ​ ​ Gaussian Elimination ​ ​ Translating between coordinate systems Translate to your own basic vectorsPerform linear transformationInverse translate basis vectors back to originalA⁻¹MA ​ Identity matrix Multiplying any matrix by the identity matrix results in the same matrix. 1 0 00 1 00 0 1 Kernel ​ ​ ​ ​ ​

### Unit Vector — a vector whose length =1

$u = v/||v||$

Tips

• Do one transformation at a time from right to left