Introduction to Linear Algebra Chapter 1 Introduction to Vectors

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The hear of linear algebra is in two operations -- both with vectors. We add vectors to get $\boldsymbol{v}$ + $\boldsymbol{w}$. We multiply them by numbers $c$ and $d$ to get $c\boldsymbol{v}$ and $d\boldsymbol{w}$. Combining those two operations(adding $c\boldsymbol{v}$ to $d\boldsymbol{w}$) gives the linear combination $c\boldsymbol{v}$ + $d\boldsymbol{w}$.

$c\boldsymbol{v}$ + $d\boldsymbol{w}$ = $c\begin{bmatrix}1\\ 1\end{bmatrix}$ + $d\begin{bmatrix}2\\3\end{bmatrix}$ = $\begin{bmatrix}c + 2d\\c + 3d\end{bmatrix}$

Central ideas of chapter 1:

Sections 1.1 Vectors addition $\boldsymbol{v} + \boldsymbol{w}$ and linear combinations $c\boldsymbol{v}$ + $d\boldsymbol{w}$

Sections 1.2 The dot product $\boldsymbol{v} \cdot \boldsymbol{w}$ of two vectors and the length $\lVert \boldsymbol{v} \rVert = \sqrt{\boldsymbol{v} \cdot \boldsymbol{v}}$

Sections 1.3 Matrics $A$, linear equations $Ax = b$, solution $x = A^{-1}b$.

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