The video introduces the concept of a linear combination in linear algebra. A linear combination of vectors involves adding them with arbitrary scaling factors. The example uses vectors \( \mathbf{a} = \begin{bmatrix} 1 \\ 2 \end{bmatrix} \) and \( \mathbf{b} = \begin{bmatrix} 0 \\ 3 \end{bmatrix} \). It demonstrates various linear combinations and shows how the span of these vectors covers all of \( \mathbb{R}^2 \). The concept of a basis for \( \mathbb{R}^2 \) is introduced, with the unit vectors \( \mathbf{i} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \) and \( \mathbf{j} = \begin{bmatrix} 0 \\ 1 \end{bmatrix} \) being highlighted as a basis for \( \mathbb{R}^2 \).
Sure, here are the key facts extracted from the text:
1. Linear combination involves adding vectors with arbitrary scaling.
2. Vectors v1, v2, ..., vn can be combined in linear combinations.
3. Linear combinations can include scaling vectors by constants (c1 to cn).
4. A linear combination of vectors v1, v2, ..., vn is represented as v1 + v2 + ... + vn, scaled by c1 to cn.
5. Concrete examples of linear combinations were provided with vectors a and b.
6. The zero vector (0 0) can be represented by a linear combination with appropriate constants.
7. The span of vectors a and b covers all of R2 (two-dimensional real space).
8. The span of a single vector a represents a line.
9. The span of unit vectors i and j forms a basis for R2, where i = (1 0) and j = (0 1).
Please note that these facts were extracted based on the provided text, and some context may be missing.