[Linear Algebra] 1. Vectors

Vector is the fundamental building block of linear algebra, so it's worth making sure what exactly a vector is.

Perspective of Vector

• Physics student: vectors are arrows, defined by length and direction, pointing in space.

• Computer Science student: vectors are ordered lists of numbers.

• Mathematician student: vector can be anything where there's a sensible notion of 1). adding two vectors and 2). multiplying a vector by a number.
  This view generalizes previous views(Physics and CS). (This hints the fact that the idea of vector addition and scalar multiplication will play an important role throughout linear algebra.)

 

When introducing a new concept about vectors, 1). think vector as arrow inside of coordinate system, like XY-plane, where tails of arrows are sitting at the origin. 2). then translate it over to the list of numbers, which are coordinate of arrow(vector). The coordinate of arrow(vector) is a pair of numbers. First number tells how far along $x$-axis and second number tells how far along $y$-axis. Every pair of list give one and only one arrow.

Vector addition & Scalar multiplication

Every topic in linear algebra will revolve around two operations. (called vector addition and scalar multiplication, respectively)

Vector addition

To add two vectors, $\mathbf{\vec{v}}$ and $\mathbf{\vec{w}}$, 1). move $\mathbf{\vec{w}}$ to $\mathbf{\vec{v}}$ tail. 2). then draw vector from origin to tip of moved $\mathbf{\vec{w}}$, that is sum of these vectors.

Geometrically

This reasonable? Think each vector as certain movement, a step with a certain distance and direction in space. First take $\mathbf{\vec{v}}$ movement, then take a $\mathbf{\vec{w}}$ movement. The overall effect is same as sum of those two vectors.

 

Numerically

The first vector $\mathbf{\vec{v}}$ has coordinate $\begin{bmatrix} v_1 \\ v_2 \end{bmatrix}$ and second vector $\mathbf{\vec{w}}$ has coordinate $\begin{bmatrix} w_1 \\ w_2 \end{bmatrix}$. Vector sum is four-step operation: 1). walk $v_1$ to $x$-axis, 2). walk $v_2$ to $y$-axis, 3). walk $w_1$ to $x$-axis, then 4). walk $w_2$ to $y$-axis. Re-organizing those steps, first do all of horizontal($x$-axis) motion, then do all of the vertical($y$-axis) motion. So vector addition in list-of-numbers perspective looks like matching up their terms and adding each one together.

$$
\mathbf{\vec{v}} + \mathbf{\vec{w}} =
\begin{bmatrix}v_1 \\ v_2 \end{bmatrix} + \begin{bmatrix} w_1 \\ w_2 \end{bmatrix} =
\begin{bmatrix} v_1 + w_1 \\ v_2 + w_2 \end{bmatrix}
$$

Scalar multiplication

Geometrically

Multiplying vector, $\mathbf{\vec{v}}$ by number, $\lambda$, is stretching or squishing or sometimes reversing the direction of a vector, $\mathbf{\vec{v}}$, by a number, $\lambda$.

Process of stretching or squishing or sometimes reversing the direction of a vector is called scaling, and number, $\lambda$, is called scalar.

 

Numerically

Multiplying a given vector, $\mathbf{\vec{v}}$, by a scalar, $\lambda$, means multiplying each one of those components by that scalar, $\lambda$.

$$
\lambda \begin{bmatrix} v_1 \\ v_2 \end{bmatrix} = \begin{bmatrix} \lambda v_1 \\ \lambda v_2 \end{bmatrix}
$$

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Linear algebra topics tend to revolve around these two fundamental operations: vector addition, and scalar multiplication. In the last chapter of this series, think about why the mathematician thinks only about these operations.

The usefulness of linear algebra is that if we view vectors as arrows, it can be interpreted as numerical representation. Also if we view vectors are list-of-numbers, it can be interpreted as geometric representation.