How to visualise bilinear form and inner products?

[mathmo94]

Hi I'm taking abstract linear algebra course and having trouble visualising bilinear form and inner products. I can visualise vector spaces, span, dimensions etc but haven't managed to figure out how to visualise this yet. Could someone please explain it to me in a visual way?
I can't understand mathematics when I can't visualise it.

 

[Office_Shredder]

Inner products are just abstracted versions of the dot product,which is really a way of measuring angles and projections. If u and v are unit vectors, then $\left< u, v \right> = \cos(\theta)$ where theta is the angle between u and v. It also equals the length of the projection of u onto v. If they are not unit vectors then you need to scale appropriately.

 

[pasmith]

Inner products are just abstracted versions of the dot product,which is really a way of measuring angles and projections. If u and v are unit vectors, then $\left< u, v \right> = \cos(\theta)$ where theta is the angle between u and v. It also equals the length of the projection of u onto v. If they are not unit vectors then you need to scale appropriately.

For completeness, the norm with respect to which $u$ and $v$ are unit vectors is
$$\|u\| = (\langle u, u \rangle)^{1/2}$$

 

[economicsnerd]

The less detailed version of the correct explanations above:

An inner product yields a notion of length for vectors (where $||x||$ denotes the length of $x$), which agrees quite well with our intuitions about length. For any pair of vectors $x,y$, the number $\langle x,y\rangle$ has size between zero and $||x||\cdot||y||$. If it has size $\approx||x||\cdot||y||$, then $x$ and $y$ are approximately parallel. If it's $\approx 0$, then $x$ and $y$ are approximately perpendicular. If it's a positive number, then $x$ and $y$ make an acute angle (from the origin). If it's a negative number, then $x$ and $y$ make an obtuse angle (from the origin).

 

[blue_raver22]

I completely understand the need to have a visual understanding of mathematical concepts. Bilinear forms and inner products are important tools in linear algebra, and they can be visualized in a few different ways.

Firstly, let's define what a bilinear form and inner product are. A bilinear form is a function that takes in two vectors and returns a scalar value. It is linear in both of its arguments, meaning that it follows the properties of linearity: f(x + y) = f(x) + f(y) and f(ax) = af(x). An inner product is a special type of bilinear form that is also symmetric, meaning that f(x,y) = f(y,x).

Now, to visualize these concepts, let's start with a simple example. Imagine two vectors, u and v, in a two-dimensional space. These vectors can be represented as arrows pointing in a certain direction and with a certain magnitude. The bilinear form between these two vectors can be visualized as the area of the parallelogram formed by extending u and v.

Next, let's consider the inner product between these two vectors. This can be visualized as the angle between the two vectors. If the angle is acute, the inner product will be positive, and if it is obtuse, the inner product will be negative. If the angle is 90 degrees, the inner product will be zero.

Another way to visualize inner products is by using the dot product. The dot product between two vectors is equal to the product of their magnitudes multiplied by the cosine of the angle between them. This can be thought of as projecting one vector onto the other and then multiplying their lengths. In this visualization, the inner product is the length of the projection vector.

In higher dimensions, the visualization becomes more complex, but the same principles apply. The bilinear form can be thought of as the volume of the parallelepiped formed by extending the vectors, and the inner product can be visualized as the angle between the vectors or as the projection of one vector onto the other.

I hope these visualizations help you to better understand bilinear forms and inner products. Remember, practice and persistence are key in understanding mathematical concepts, and with time, you will develop a strong visual understanding of these concepts.