SUMMARY
Every vector in $\mathbb{R}^2$ can be expressed in the form $au + bv$, where $u = (0, 1)$ and $v = (1, 0)$, with $a, b \in \mathbb{R}$. This representation holds true under the condition that the vectors $u = (u_1, u_2)$ and $v = (v_1, v_2)$ are linearly independent, which is confirmed when the determinant $u_1v_2 - v_1u_2 \neq 0$. If the vectors are orthogonal, the representation remains valid, while non-orthogonal vectors also satisfy this condition as long as they are not collinear.
PREREQUISITES
- Understanding of linear combinations in vector spaces
- Familiarity with the concept of linear independence
- Knowledge of determinants in linear algebra
- Basic proficiency in $\mathbb{R}^2$ vector representation
NEXT STEPS
- Study linear independence and its implications in vector spaces
- Learn about determinants and their role in solving linear systems
- Explore the concept of vector spaces and bases in linear algebra
- Investigate the geometric interpretation of vectors in $\mathbb{R}^2$
USEFUL FOR
Students of linear algebra, mathematicians, and anyone interested in understanding vector representations in two-dimensional space.