Math
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Linear independenceMath/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 spaceMath/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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Dot productMath/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 $$\alpha =a\cdot b=\..
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Invertible matrix and Pseudo-inverseMath/Linear algebra 2019. 10. 8. 15:36
1. Overview $$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..