Line 18: | Line 18: | ||
Standard Basis: | Standard Basis: | ||
* the identity $I_d$, | * the identity $I_d$, | ||
− | * e.g. for | + | * e.g. for <math>\mathbb R^3</math>, <math>\mathbf e_1 = \begin{bmatrix} |
1 \\ 0 \\ 0 | 1 \\ 0 \\ 0 | ||
− | \end{bmatrix} | + | \end{bmatrix}</math>, <math>\mathbf e_2 = \begin{bmatrix} |
0 \\ 1 \\ 0 | 0 \\ 1 \\ 0 | ||
− | \end{bmatrix} | + | \end{bmatrix}</math>, <math>\mathbf e_3 = \begin{bmatrix} |
0 \\ 0 \\ 1 | 0 \\ 0 \\ 1 | ||
− | \end{bmatrix} | + | \end{bmatrix}</math> |
Non-Example: | Non-Example: | ||
− | * | + | * <math>\begin{bmatrix} |
1 \\ 1 \\ 2 | 1 \\ 1 \\ 2 | ||
− | \end{bmatrix} | + | \end{bmatrix}</math>, <math>\begin{bmatrix} |
2 \\ 2 \\ 5 | 2 \\ 2 \\ 5 | ||
− | \end{bmatrix} | + | \end{bmatrix}</math>, linearly independent, but don't span $\mathbb R^3$ |
− | * | + | * <math>\begin{bmatrix} |
1 \\ 1 \\ 2 | 1 \\ 1 \\ 2 | ||
− | \end{bmatrix} | + | \end{bmatrix}</math>, <math>\begin{bmatrix} |
2 \\ 2 \\ 5 | 2 \\ 2 \\ 5 | ||
− | \end{bmatrix} | + | \end{bmatrix}</math>, <math>\begin{bmatrix} |
3 \\ 3 \\ 7 | 3 \\ 3 \\ 7 | ||
− | \end{bmatrix} | + | \end{bmatrix}</math>, the 3rd vector is a linear combination of first 2 |
* the first case is 2 vectors on a plane, and 2nd is 3 vectors on a plane | * the first case is 2 vectors on a plane, and 2nd is 3 vectors on a plane | ||
* http://habrastorage.org/files/328/4c6/a34/3284c6a346ff491e8ac295ec82ea1f91.png | * http://habrastorage.org/files/328/4c6/a34/3284c6a346ff491e8ac295ec82ea1f91.png | ||
Line 45: | Line 45: | ||
Another example: | Another example: | ||
− | * | + | * <math>\begin{bmatrix} |
1 \\ 1 \\ 2 | 1 \\ 1 \\ 2 | ||
− | \end{bmatrix} | + | \end{bmatrix}</math>, <math>\begin{bmatrix} |
2 \\ 2 \\ 5 | 2 \\ 2 \\ 5 | ||
− | \end{bmatrix} | + | \end{bmatrix}</math>, <math>\begin{bmatrix} |
3 \\ 1 \\ 8 | 3 \\ 1 \\ 8 | ||
− | \end{bmatrix} | + | \end{bmatrix}</math> |
* if we put the vectors as columns of a matrix, then the rank should be equal to the number of vectors | * if we put the vectors as columns of a matrix, then the rank should be equal to the number of vectors | ||
− | * | + | * <math>\begin{bmatrix} |
1 & 2 & 3\\ | 1 & 2 & 3\\ | ||
1 & 2 & 1\\ | 1 & 2 & 1\\ | ||
2 & 5 & 8 \\ | 2 & 5 & 8 \\ | ||
− | \end{bmatrix} | + | \end{bmatrix}</math>, rank is 3 |
In Linear Algebra, basis is a set of linearly independent vectors $\mathbf v_1, ..., \mathbf v_n$
Vectors $\mathbf v_1, ..., \mathbf v_l$ span a (sub)space $\iff$ this space consists of all possible linear combinations of these vectors
Basis of a vector space is a sequence of vectors $\mathbf v_1, \mathbf v_2, ..., \mathbf v_d$ that
Standard Basis:
Non-Example:
Another example:
Also,
Every space has some basis, and each basis of this space has the same number of vectors. The number of vectors in the basis is the dimension of this space.