Calculate inner product of two vectors

Hint: $\;e_1 \cdot v = e_1 \cdot (2e_1 - 3e_2 + 4e_3) = 2 |e_1|^2- 3 e_1 \cdot e_2 + 4 e_1 \cdot e_3 = 2 \cdot 3^2 - 3 \cdot (-6)+4 \cdot 4\,$. Now do the same for $\,u \cdot v\,$ instead of $\,e_1 \cdot v\,$.


HINT

The inner product in $\mathbb{R}^n$ has these three properties:

$$u \bullet v=v \bullet u$$

$$(v_1+v_2) \bullet u=u \bullet (v_1+v_2)=(u \bullet v_1)+(u \bullet v_2)$$

$$a(u \bullet v)=(au) \bullet v=u \bullet (av),\forall a \in \mathbb{R}$$

Use them and you will get the result.


Hint:

The inner product is a bilinear form, i.e. its linear in each variable and takes values in the base field $\mathbf R$.

Tags:

Linear Algebra

Inner Products

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