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

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Linear Transformation

Conditions: 1. Origin remains fixed

2. All lines remain straight

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Linear Combination

  • Sum of two vectors

  • Tip-to-tail method

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Span

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

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Linearly dependent

Remove one point and there is no change

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Scalars

numbers

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Composition

Product of two matrices in a single action

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Determinant

Scaling factor

  • if negative, orientation is flipped

  • det(mโ‚mโ‚‚) = det(mโ‚)det(mโ‚‚)

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Dot product

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

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Cross product

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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.

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Eigenvalue

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Eigenvector

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Rank

Rank 1: Single line

Rank : 2D Plane

Rank 3: 3D plane

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Duality

Natural but surprising correspondence

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Cramer's rule

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Gaussian Elimination

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Translating between coordinate systems

  1. Translate to your own basic vectors

  2. Perform linear transformation

  3. Inverse translate basis vectors back to original

AโปยนMA

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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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Unit Vector โ€” a vector whose length =1

u=v/โˆฃโˆฃvโˆฃโˆฃu = v/||v||

Tips

  • Do one transformation at a time from right to left

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