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

Checking Linearity For the mapping defined in each of Problems 1-16, determine whether $a$ not it is a linear transformation. $T: \mathbf{M}_{22} \rightarrow \mathbb{R}, \quad T\left[\begin{array}{ll}a & b \\ c & d\end{array}\right]=\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|$

Checking Linearity For the mapping defined in each of Problems 1-16, determine whether $a$ not it is a linear transformation.
$T: \mathbf{M}_{22} \rightarrow \mathbb{R}, \quad T\left[\begin{array}{ll}a & b \\ c &
d\end{array}\right]=\left|\begin{array}{ll}a & b \\ c & d\end{array}\right|$

Key Concepts

Linear Transformation

A linear transformation is a function between vector spaces that preserves the operations of addition and scalar multiplication. This means that for any vectors u and v, and any scalar c, the transformation T must satisfy T(u + v) = T(u) + T(v) and T(c*u) = c*T(u). These properties ensure that the structure of the vector space is maintained under the mapping.

Determinant Function

The determinant is a scalar function that assigns a single value to a square matrix, encapsulating key properties such as scaling of volume and invertibility. However, the determinant does not generally preserve addition or scalar multiplication, and thus it does not satisfy the criteria for a linear transformation. This non-linearity is evident when one observes that the determinant of the sum of matrices is not the sum of their determinants.

Transcript

00:01 Hello there.

00:02 In the following exercise, we have a transformation from the space of matrices 2 to the reels.

00:08 The transformation, we have in this case, we're going to have two transformations.

00:12 Let's read the first one.

00:14 The first one is defined as follow.

00:16 We pick a matrix a, b, c and d, and we obtain this sum of terms on the right.

00:24 So we need to show if this transformation is linear or not, and in case that is linear, we need to find the kernel.

00:32 Help to show that this transformation is linear.

00:35 Well, all the linear transformation satisfy two axioms.

00:38 The first one is that the transformation of the sum of the sum of the transformation of these two terms, and that if we take, if we multiply any scalar, in this case alpha, it's equal to multiply the scalar to the transformer vector.

01:02 So, as you can observe, we need to pick two elements in the domain of this transformation.

01:09 In this case, our domain is the space of matrices 2x2.

01:14 So let's pick two vectors in the same form as it's defined here in the transformation.

01:21 So let's pick the matrix a1, b1, c1, d1, and the matrix a2, b2, c2, and d2 these two are matrices in m2 2 2 so we need to show and i'm going to give some labels so let's call these the matrix a 1 and the matrix a 2 so what we need to show is a t of a 1 plus a 2 it's equals to t a 1 plus t a 2 okay so let's let's see.

02:19 Let's take the left -hand side of this expression.

02:23 So we have t of a1 plus a2.

02:28 There's no more than applying the transformation to the matrix a1 plus a2, b1 plus b2, c1 plus c2, and d1 plus d2.

02:52 If we follow the definition of the transformation, you're going to obtain that this equals to 3 times a1 plus a2 minus 4 times b1 plus b2 plus c1 plus c2 and minus the 1 minus d2.

03:21 Now what i want you to observe is that we can separate the terms that has the subscript 1.

03:30 Means that this is equal to 3a1 minus 4b1 plus c1 and minus d1.

03:39 And on the other hand, we have this 3a2 minus 4 b2 plus c2 minus d2.

03:49 And what happened is that this is the definition of applying t to the first matrix a1.

03:57 And this part here corresponds to applying t to the second matrix.

04:01 Matrix a2.

04:04 So the first property of these linear transformations is satisfied.

04:10 That's great...