Find a linear function L: ℝ^2 →ℝ such that L( 1 1 )=2 and L( 1 -1 )=3. Is it

step 1 passing through two points, the point that's called v1 and v2. So the first point is minus 1, 4,

Find a linear function L: ℝ^2 →ℝ such that L( 1 ...

Key Concepts

Uniqueness of Linear Transformations

The uniqueness of a linear transformation is guaranteed when its values on a basis of the domain are specified. Since any vector is a combination of basis elements, knowing the transformation's effect on each basis vector uniquely determines its effect on every vector in the space. This concept is central to understanding conditions under which a linear function is uniquely defined.

Solving Systems of Linear Equations

Determining the parameters of a linear function often involves setting up and solving systems of linear equations based on the function's outputs for given inputs. This approach is essential when matching the function's values with specific conditions, as it allows one to solve for the unknown coefficients in the function's representation.

Linear Functions

A linear function (or linear transformation) is a mapping between vector spaces that preserves vector addition and scalar multiplication. This concept is fundamental in linear algebra, as it ensures that the structure of the space is maintained under the function, making it possible to predict the behavior of the transformation on any vector by knowing its effect on a set of basis vectors.

Basis Representation

In a finite-dimensional vector space, any vector can be expressed as a linear combination of basis vectors. A linear function is completely determined by its values on a basis. Thus, if you know the output of the linear function on a basis, you can compute its value on any vector. This is crucial for understanding how to construct and uniquely identify a specific linear transformation.

Transcript

00:02 Passing through two points, the point that's called v1 and v2.

00:05 So the first point is minus 1, 4, and the second point is 3 -1.

00:13 So geometrically thinking, there is some point here v1, there is another here v2, and we got here a line, passing through these two points.

00:24 So the point of this exercise actually is they're asking for a functional f and a real value d, such that this line is equals to that functional.

00:38 So that means, that implies that we need to find that first this f and then what is the value of this d.

00:46 So just to remember that this functional fd, just for notation, means the set of all the points, in this case xy on r2, such that this function evaluated at eighty, x y is equal to d okay so this is the formal definition of this linear functional so let's start by considering just the line that pass that is perpendicular to these two points so now what we're going to do actually is construct here we got these these two points here we are going to define a vector this vector will be defined by v2 minus b1 and we want to find a normal vector to this to this vector v2 minus v1 because this is the usual way to construct an hyperplane in this case it's just in r2 so just to remember that the the the hyperplane center on the origin is given by this okay so this is the equation where x is a vector so if we write it in this way if n is equal to n1 and n2 then this equation here is equals to n1 n2 that product x1 x2 okay so this is just a line that that pass through the center of the through the origin of this of the of the of our two okay so this equation here in green represents a line that passed through the origin and that is parallel to the line that passed through the points b1 and b2 so it's parallel but is center at the origin okay so now so that's the first thing that we need we need to do is find that n that perpendicular vector.

03:12 So how to do that? well, as i mentioned, first we need to compute v2 minus v1.

03:18 This is equal to 3 -1 minus minus 1 4 and this is equal to 4 minus 3.

03:32 Okay, so this is the the first vector b1, b2 and now we now we need to find a vector that is perpendicular to this one.

03:44 We're going to call n and to find the equation of the line that is parallel to this vector here.

03:53 Okay, so that's the idea.

04:00 So how to find an orthogonal vector? well, we are going to use the definition of the two vectors are orthogonal.

04:08 That means that the dot product is equal to zero.

04:11 So 4 minus 3 is equal to 0.

04:17 Okay, so we got this...

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