https://datascienceschool.net/view-notebook/41138b1b54454caf819d9288d26560c7/ ================================================================================ Linearly dependent - N number of vectors - $$$x_1,x_2,\cdots,x_N$$$ - when following is true: - $$$c_1 x_1 + c_2 x_2 + \cdots + c_N x_N = 0 $$$ - c set which satisfies above equation exists - exclude this case: $$$c_1=c_2=\codts=c_N=0$$$ Linearly independent - N number of vectors - $$$x_1,x_2,\cdots,x_N$$$ - when following is true: - $$$c_1 x_1 + c_2 x_2 + \cdots + c_N x_N = 0 $$$ - c set which satisfies above equation doesn't exist - exclude this case: $$$c_1=c_2=\codts=c_N=0$$$ ================================================================================ - Suppose there is N number of vectors - Perform linear combination with N number of vectors - Result from linear combination is another vector - All vectors from linear combination are "vector space V" - Vector space V's dimension is N (N number of vectors) - N number of vectors are "basis vector v" $$$V = \{ c_1x_1 + \cdots + c_Nx_N \; \vert \; c_1, \ldots, c_N \in \mathbf{R} \}$$$ ================================================================================