https://datascienceschool.net/view-notebook/41138b1b54454caf819d9288d26560c7/
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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: $$$c_1=c_2=\codts=c_N=0$$$
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- 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"
$$$V = \{ c_1x_1 + \cdots + c_Nx_N \; \vert \; c_1, \ldots, c_N \in \mathbf{R} \}$$$
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