02_005_Basis_vector_Vector_space.html =================================================================================== You can think of all kind of (n,1) matrix You also think of "vector set" which can express all kind of (n,1) matrix In this situation, you can say "(n,1) above vector set" is "basis vector set" "Arbitrary one vector" can be expressed like it gets spanned into (n,1) vector space via basis vector set Suppose there are N number of basis vectors: $$$\{v_{i}\}_{1\leq i \leq N}$$$ You can create "arbitrary vector x" by using following formular : $$$x = \sum\limits_{i=1}^{N}c_{i}v_{i}$$$ "Vector x" is located in "n-D basis vector's space" as one point In other words, vector x is created via linear combination of basis vector set v -------------------------------------------------- Basis vectors $$$v_{1}, v_{2}, v_{3}$$$ in 3-D space $$$v_{1} = \begin{bmatrix} 1\\0\\0 \end{bmatrix}$$$ $$$v_{2} = \begin{bmatrix} 0\\1\\0 \end{bmatrix}$$$ $$$v_{3} = \begin{bmatrix} 0\\0\\1 \end{bmatrix}$$$ $$$\begin{bmatrix} 3\\6\\9 \end{bmatrix} = 3\times \begin{bmatrix} 1\\0\\0 \end{bmatrix} 6\times \begin{bmatrix} 0\\1\\0 \end{bmatrix} 9\times \begin{bmatrix} 0\\0\\1 \end{bmatrix}$$$ ====================================================================== If "arbitrary one vector a" can be expressed via linear combination of vector set $$$\{u_{i}\}$$$ like this: $$$a = a_{1}u_{1}+a_{2}u_{2}+...+a_{N}u_{N}$$$ vector set $$$\{u_{1},u_{2},...,u_{N}\}$$$ forms "basis" with respect to "vector space" and "vector a" You can say coefficient set $$$\{a_{1},a_{2},...,a_{N}\}$$$ is element of vector a according to basis vector set $$$\{u_{i}\}$$$ That is, to form vector a, $$$u_{1}$$$ exists in size of $$$a_{1}$$$, $$$u_{2}$$$ exists in size of $$$a_{2}$$$, ... And, to form basis, linearly independent of $$$\{u_{i}\}$$$ is necessary and sufficient condition Orthogonal : $$$u_{i}^{T}u_{j} \neq 0$$$, i=j $$$u_{i}^{T}u_{j} = 0$$$, $$$i\neq j$$$ Orthonormal : $$$u_{i}^{T}u_{j} = 1$$$, i=j $$$u_{i}^{T}u_{j} = 0$$$, $$$i\neq j$$$ ====================================================================== Summary "Arbitrary all vectors x" can be expressed via "linear combination" of "basis vectors" which form space where all vectors x exist $$$x = \sum\limits_{i=1}^{N}c_{i}v_{i}$$$ In that situation, basis vectors which you can use for above task (linear combination to create all vectors x) should be linearly independent to each other In other words, basis vectors should be orthonormal to each other In other words, basis vectors should have 90 degree to all others and have 1 length