A Bi Form Calculator

Bilinear Form Calculator

Enter the elements of the 2×2 matrix A and the components of the 2-dimensional vectors u and v to calculate the bilinear form B(u, v) = u^TAv.

Understanding the Bilinear Form Calculator

This calculator helps you compute the value of a bilinear form, a fundamental concept in linear algebra. A bilinear form is a function that takes two vectors as input and produces a scalar (a single number) as output, satisfying linearity in each argument separately.

What is a Bilinear Form?

In mathematics, particularly in linear algebra, a bilinear form is a mapping B: V × V → F, where V is a vector space over a field F (e.g., real numbers). It's called "bilinear" because it is linear with respect to each of its two arguments when the other argument is held constant. This means for any vectors u, v, w in V and scalars a, b in F:

1. B(au + bw, v) = aB(u, v) + bB(w, v) (linearity in the first argument)
2. B(u, av + bw) = aB(u, v) + bB(u, w) (linearity in the second argument)

A common example of a bilinear form is the dot product (or scalar product) in Euclidean space, which is also a symmetric bilinear form.

Matrix Representation of a Bilinear Form

For finite-dimensional vector spaces, any bilinear form can be represented by a matrix. If V is an n-dimensional vector space, and u and v are column vectors representing elements of V with respect to some basis, then the bilinear form B(u, v) can be expressed as:

B(u, v) = uTAv

Where:

• uT is the transpose of vector u (a row vector).
• A is an n x n matrix, known as the matrix of the bilinear form.
• v is a column vector.

This calculator specifically handles 2-dimensional vectors and a 2×2 matrix. For u = [u1, u2]T, v = [v1, v2]T, and A = [[a11, a12], [a21, a22]], the calculation expands to:

B(u, v) = u1(a11v1 + a12v2) + u2(a21v1 + a22v2)

This formula is what the calculator uses to determine the scalar output.

How to Use This Calculator

To use the Bilinear Form Calculator, simply input the numerical values for the elements of your 2×2 matrix A and the components of your two 2-dimensional vectors, u and v:

1. Matrix A Elements: Enter the values for a11, a12, a21, and a22. These define the specific bilinear form.
2. Vector u Components: Input the values for u1 and u2, which are the components of your first vector.
3. Vector v Components: Input the values for v1 and v2, which are the components of your second vector.
4. Click the "Calculate Bilinear Form" button. The result, a single scalar value, will be displayed below.

Example Calculation

Let's consider an example:

• Matrix A: [[1, 2], [3, 4]] (so a11=1, a12=2, a21=3, a22=4)
• Vector u: [5, 6]T (so u1=5, u2=6)
• Vector v: [7, 8]T (so v1=7, v2=8)

Using the formula B(u, v) = u1(a11v1 + a12v2) + u2(a21v1 + a22v2):

First, calculate Av:

Av = [[1, 2], [3, 4]] * [7; 8] = [ (1*7 + 2*8); (3*7 + 4*8) ] = [ (7 + 16); (21 + 32) ] = [23; 53]

Now, calculate uT(Av):

uTAv = [5, 6] * [23; 53] = 5*23 + 6*53 = 115 + 318 = 433

So, the bilinear form B(u, v) for these inputs is 433. You can verify this result using the calculator above.