Demyank's Tlog Demyank's Tlog

Math/Linear algebra

• Linear independence

  Math/Linear algebra 2019. 10. 10. 14:42

  1. Overview 2. Description 2.1 Definition A set of M vectors is independent if each vector points in a geometric dimension not reachable using other vectors in the set. Any set of M>N vectors in $\mathbb{R}^{N}$ is dependent Any set of $M\leq N$ vectors in $\mathbb{R}^{N}$ could be independent 2.2 How to determine whether a set is independent Step 1: Count vectors and compare with $R^{N}$ Step 2..

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• Difference between space, subspace, subset, span, and ambient space

  Math/Linear algebra 2019. 10. 10. 10:06

  1. Overview 2. Description 2.1 Space 2.2 Subspace Be closed under addition and scalar multiplication Contain the zero vector $\forall \boldsymbol{v},\boldsymbol{w}\in V$; $\forall \lambda ,\alpha \in \mathbb{R}$; $\lambda\boldsymbol{v} +\alpha\boldsymbol{w} \in V$ 2.3 Subset 2.3.1 Definition 2.3.2 How to distinguish subset and subspace Step 1: Determine whether the origin is in the set Step 2: ..

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• Field

  Math/Linear algebra 2019. 10. 10. 03:12

  1. Overview 2. Description 3. References https://en.wikipedia.org/wiki/Field_(mathematics)

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• Diagonalizable matrix

  Math/Linear algebra 2019. 10. 9. 20:04

  1. Overview 2. Description 3. Example $$diag\left ( \begin{pmatrix} 1 & -1 & 8\\ -1 & -2 & 4\\ 0 & 3 & 5 \end{pmatrix} \right )$$ 3. References https://en.wikipedia.org/wiki/Diagonalizable_matrix

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• Cross-product

  Math/Linear algebra 2019. 10. 9. 19:59

  1. Overview 2. Description 2.1 Definition 2.2 Cross-product in geometry 3. References https://en.wikipedia.org/wiki/Cross_product#targetText=The%20cross%20product%20a%20%C3%97,is%20defined%20by%20the%20formula

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• Dot product

  Math/Linear algebra 2019. 10. 9. 00:20

  1. Overview In mathematics, the dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers (usually coordinate vectors) and returns a single number. 2. Description 2.1 Algebraic definition Algebraically, the dot product is the sum of the products of the corresponding entries of the two sequences of numbers. 2.2 Algebraic Notation $$\alpha =a\cdot b=\..

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• Invertible matrix and Pseudo-inverse

  Math/Linear algebra 2019. 10. 8. 15:36

  1. Overview $$AB=BA=I_{n}$$ 2. Description 2.1 Pre-requirement A matrix is invertible if it is Square matrix Full rank Non-zero determinant 2.2 Properties Let $A$ be a square n by n matrix over a field $K$ (for example the field $R$ of real numbers). The following statements are equivalent for any given matrix are either all true or all false: A is invertible, that is, A has an inverse, is nonsi..

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• Gauss Elimination and Gauss-Jordan Elimination

  Math/Linear algebra 2019. 10. 7. 18:23

  1. Overview Row reduction is the process of performing row operations to transform any matrix into (reduced) row echelon form. In reduced row echelon form, each successive row of the matrix has less dependencies than the previous, so solving systems of equations is a much easier task. The idea behind row reduction is to convert the matrix into an "equivalent" version in order to simplify certain..

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