## Euclidean space geometry: vector operations

The combination of these words may sound frightening. In fact, if you want to succeed with matrix algebra, you need to start drawing inspiration from geometry as early as possible.

### Sum of vectors

**Definition**. The set of all -dimensional vectors with is denoted and is called a **Euclidean space**.

is a plane. The space we live in is Our intuition doesn't work in dimensions higher than 3 but most facts we observe in real life on the plane and in the 3-dimensional space have direct analogs in higher dimensions. Keep in mind that can be called a *vector* or a *point* in depending on the context. When we think of it as a vector, we associate with it an arrow that starts at the **origin** and ends at the point

Careful inspection shows that the **sum of two vectors** is found using the **parallelogram rule** in Figure 1. The rule itself comes from physics: if two forces are applied to a point, their resultant force is found by the parallelogram rule. Whatever works in real life is guaranteed to work in Math.

**Exercise 1**. Let **unit vector** of the

This seemingly innocuous exercise leads to profound ideas, to be considered later. The answer for the last question is that the sums

### Multiplication of a vector by a number

If **scaling** or **multiplication by a number**). Scaling

**Exercise 2**. Take a nonzero vector

The answer is that

The expression **linear combination** of vectors

**Exercise 3**. Let

Verbally, this is the set of all linear combinations

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