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 |
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Span |
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Linearly dependent | Remove one point and there is no change | |
Scalars | numbers | |
Composition | Product of two matrices in a single action | |
Determinant | Scaling factor
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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
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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 |
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 | | |
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Tips
Do one transformation at a time from right to left