Note that the last term of the equation shows a matrix multiplication of a 1-times-2 and a 2-times-1 matrix. An interpretation of inner products is not that obvious. An inner product basically provides a meausre of how similar the two input vectors are. A geometric illustration of dot products is explained by 3Blue1Brown.
While watching bear in mind that an inner product is a symmetric bilinear form. If you are interested in the duality of vector spaces, refer to this article in German. There is also another nice video of applications of inner products by Zach Star :.
It might also help to learn more about norms and metrics. Knowing that inner products, norms and metrics are closely related to each other means that understanding one concept provides also heuristic information about the other concept.
Functions closely related to inner products are so-called norms. Norms are specific functions that can be interpreted as a distance function between a vector and the origin.
Definition 3. A function with is called norm , and, is called normed space if for all the following holds true:. Properties ii and iii make only sense over a vector space. The former property is stating how the scalar product of a vector and a field element needs to behave to be a valid norm function. Example 3. It is always positive and only zero if and it also fulfills homogenity by definition. To see that the triangle inequality also called subadditivity holds, first bear the definition of the absolute value in mind.
If and as well as then. In addition, , as well as , hold true. Hence, by adding both inequalities we get as well as , as desired. Note that is the Euclidean inner product as defined in Example 2.
If the Euclidean norm can be interpreted as the length between the origin and the vector , then the Euclidean inner product can be interpreted as the squared norm.
Example a is actually the most important example of a norm since basically every practically important norm can be traced back to the absolute value function in some sense. The Euclidean norm of the vector equals , which is exactly the length of hypotenuse of the corresponding triangle in Figure 2. It also explains why the property iii is called triangle property and sometimes also subaditivity.
The norm is also called distance between the two real-valued vectors and. The angle between two vectors and with is defined by. Ask Question. Asked 2 years, 8 months ago. Active 8 months ago. Viewed 1k times. Improve this question. Jaigus Jaigus 1, 1 1 gold badge 14 14 silver badges 29 29 bronze badges. Add a comment. Active Oldest Votes. Improve this answer. Leandro Caniglia Leandro Caniglia Sign up or log in Sign up using Google. Sign up using Facebook.
Sign up using Email and Password. Post as a guest Name. Active 4 years, 6 months ago. Viewed 32k times. Wikipedia says: A vector can be described as a directed line segment from the origin of the Euclidean space vector tail , to a point in that space vector tip. I would very much appreciate it if somebody could clear the haze.
You can define the norm in terms of the metric, or you can define the metric in terms of the norm. It doesn't matter which order you do it in.
Perhaps my confusion stems from the fact that on the one hand, as you say, you can do it either way, on the other hand there is this hierarchy between metric spaces and normed spaces every normed space is a metric space but not the other way round.
It seems like a contradiction to me?!? There is no contradiction here. Add a comment. Active Oldest Votes. Gerry Myerson Gerry Myerson k 11 11 gold badges silver badges bronze badges. Sign up or log in Sign up using Google. Sign up using Facebook. Sign up using Email and Password.
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