> [!summary] Linear combinations of vectors are a combination of two or more vectors > >[!info]+ Read Time **⏱ 1 min** # Definition Linear combinations of [[Scalar & Vectors|vectors]] in linear algebra are when you multiply specific vectors by a scalar amount and [[Vectors Addition & Multiplication|add them together]]. For example, consider the image below with two vectors in the $i$ and $j$ directions. If we added the two vectors together, the result would be a **linear combination** of the two vectors, being ${v=\begin{bmatrix} 1 \\ 1 \end{bmatrix}}$. ![[lc_1.png | 400]] However, there are many possible results if we multiply any of these components by a scalar multiple of each vector, such that the linear combinations is as followed $v = 3i{\begin{bmatrix} 0 \\ 1 \end{bmatrix}} + 2j{\begin{bmatrix} 1 \\ 0 \end{bmatrix}}$, ![[lc_2.png|400]] # Resources <iframe width="560" height="315" src="https://www.youtube.com/embed/k7RM-ot2NWY?si=1Dr3rTsC1fOddffV" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe> <iframe width="560" height="315" src="https://www.youtube.com/embed/WJlQzgS_itI?si=qLasdCc4F9fUVl0u" title="YouTube video player" frameborder="0" allow="accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture; web-share" referrerpolicy="strict-origin-when-cross-origin" allowfullscreen></iframe> --- > ✍️ This project’s been a labour of love. > If it helped, [give Math & Matter a star](https://github.com/rajeevphysics/Obsidan-MathMatter) and let me know what you'd like to see next. ---