Checking Linearity For the mapping defined in each of Problems 1-16, determine whether a not it

Checking Linearity For the mapping defined in each of...

Key Concepts

Linear Transformation

A linear transformation is a mapping between two vector spaces that preserves the operations of vector addition and scalar multiplication. This means that for any two vectors u and v in the domain and a scalar c, a transformation T must satisfy T(u + v) = T(u) + T(v) and T(cu) = cT(u) to be considered linear.

Additivity

Additivity, or the property of being additive, means that when the linear transformation is applied to the sum of two vectors, the result is the same as adding together the images of each vector under the transformation. This condition, T(u + v) = T(u) + T(v), is essential for the mapping to qualify as a linear transformation.

Homogeneity

Homogeneity, also known as scalar multiplication preservation, is the property that scaling a vector by a scalar and then applying the transformation is equivalent to applying the transformation first and then scaling the result. Formally, this is expressed as T(cu) = cT(u) for any scalar c and vector u.

Vector Space of Polynomials

The vector space of polynomials, such as P3, consists of all polynomials up to a certain degree (in this case, degree 3) and is equipped with operations of addition and scalar multiplication. Understanding the structure of this space is key when analyzing transformations defined on it, such as those that extract or combine coefficients.

Transcript

00:01 Hi there.

00:02 In the following exercise we have a transformation from the space of polynomials of at most degree 2 to this same to itself.

00:10 Okay and we have this definition for the first part we're defining this transformation as follows.

00:19 Okay and we need to show we need to check actually if this is linear or not.

00:27 So let's see a linear transformation can satisfy two axioms.

00:32 The first one is is that the transformation of the sum of two vectors in the main space, it's equals to the sum of the transformations.

00:46 So that's what it said the first part.

00:49 And the second axioms say that if we multiply an scalar, this alpha is just a real number to a vector, it's equals to multiply this constant to the transformate vector.

01:02 So that are the two axioms that a linear transformation must sentence.

01:06 And we need to check if this transformation in particular satisfy those conditions.

01:10 So we need to consider two polynomials.

01:14 So let's consider the polynomial p1 that is equal to a0 plus a1x plus a2x squared and the polynomial 2m2 that's going to be equal to v0 plus b1x plus v2.

01:41 2x square.

01:43 Now you can observe that we can regret this transformation as follows.

01:48 Basically, this transformation say that t of px is equals to p of x plus 1.

01:57 Okay, there is nothing more than that.

02:03 So i'm writing this in this form because this makes easier to prove that it is a new transformation.

02:11 So for the first one, we need to show.

02:13 We need to show that t applied to p1 plus p2 it's equal to to apply this transformation to p1 and then to p2 so let's take the left hand side t of p1 plus p2 will be no more than p1 plus b2 apply to x plus 1 and we can distribute this basically this is equal to p1 applied to x plus 1 plus p2 apply to x plus 1 and as you can observe this is equals to t applied to p1 plus t applied to p2 great and then the second part the second part is the second axiom consists on a proof that alpha applied to p1 multiplied p1 is equal to to alpha times the transformation to the first polynomial.

03:33 So let's take the left -hand side...

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