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

Additivity and Homogeneity

These are the core properties required for a mapping to be considered linear. Additivity means that the transformation of a sum of elements equals the sum of their individual transformations, while homogeneity (or scalar multiplication preservation) means that multiplying an input by a scalar results in the output being multiplied by the same scalar. Together, these properties determine whether a given mapping is a linear transformation.

Matrix Transpose Operation

The matrix transpose is an operation that interchanges the rows and columns of a matrix. This operation is often discussed in linear algebra because it interacts with the structure of matrices in a way that can preserve or affect properties like symmetry and linearity.

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 and any scalar, the image of their sum is the sum of their images, and the image of a scalar multiple of a vector is that scalar multiplied by the image of the vector.

Vector Spaces

A vector space is a collection of elements (such as matrices, functions, or vectors) where addition and scalar multiplication are defined and satisfy certain axioms. In the context of linear transformations, both the domain and codomain are vector spaces that must support these operations for the concept of linearity to be applicable.

Transcript

00:01 So we've got this set, this transformation two going from the set of two by two matrices to the set of two by two matrices.

00:08 And it's given by sending a two by two matrix a to a plus a transpose.

00:16 Now t is linear.

00:18 Okay.

00:20 So the transformation a to a, which is just the identity, is obviously linear.

00:26 And the transformation a to a transpose is also linear.

00:31 The proof that a to a transpose is linear is just by checking, it's just by using the fact that lambda a transpose is equal to lambda times a transpose and a plus b transpose is equal to a transpose plus b transpose.

00:53 And it follows from these facts.

00:57 However, if you want to see, so this is, if you want to see this more directly, let's check directly that t is linear.

01:07 So let a and b be any 2x2 matrices.

01:14 Okay.

01:16 Then t of a plus b is equal to a plus b plus a plus b transpose, but then that's a transpose plus b transpose plus b transpose.

01:28 And then this is the same as this, which is t of a plus t of b.

01:37 Now for the second part, so this is not, so you need to check that t of a plus b is t of a plus t of b, but also we need to check that it behaves like we wanted to under scalar multiplication.

01:55 So t of lambda a is equal to lambda a plus lambda a transpose, but that's just lambda a plus lambda a transpose, which is lambda into a plus a transpose.

02:08 Goes.

02:12 Okay, so we've proven that t is linear, right? now we want to decide whether it's one -to -one onto both or neither.

02:24 So the first thing that's useful to check is the dimension of the domain and the co -domain.

02:31 Now the dimension of the domain is the same as the dimension of the co -domain because they're the same space and the dimensionality is four.

02:42 Okay.

02:43 How do i know that that? the vector space of two by two matrices is four dimensional? well, a perfectly good basis you can use is given by this.

02:55 And there's four of them.

03:07 Okay, so let's start off by figuring out what the kernel is, because that will give us information as to if it is one to one or if it is not.

03:17 So the kernel of t is just the set of all a in the domain such that t of a, such that a gets mapped to the zero element in the code domain, which is just the zero matrix, the zero two by two matrix.

03:39 So elements in this set have the property that a plus a transpose is equal to 0 -0 -0 -0 -0.

03:51 So this is the set of all anti -symmetric matrices.

03:57 Okay, so okay, let's do this directly.

04:00 So suppose that, so as entries a, b, c, d, and the transpose is ac, b, d, and we want this to equal 0 ,000, so that gives us that a is equal to d is equal to 0, and that b is equal to negative c.

04:17 So, the kernel of t is the set 0 little t minus t, such that t is a real number.

04:27 So it's a one -dimensional subspace, of m2r.

04:32 The kernel is not zero.

04:35 It's not just the zero matrix.

04:36 So we know it is not t is not one to one.

04:43 And therefore, it is not onto because, well, because we're going from the same, because the domain and the co -domain have the same dimension, right? the only way for t to be one to one and onto, well, okay, so since t is not one to one, then it can't be onto, because the domain and the code domain have the same dimension.

05:19 And in fact, yeah, we can see this, so i'll use the rank nullity theorem in a second, but let's first, let's first figure out a basis for kernel of t.

05:32 So basis for kernel of t is just given by 0 -1 -0 -0.

05:38 All right, this is a linearly, dependent set and it certainly spans this subspace right here.

05:50 Okay, so let's talk about the image of t.

05:54 So, t is not onto, okay, so the image of t is not equal to m2r.

06:01 How do i know this? well, from the rank nullity theorem, we've got that the rank of t plus the nullity of t must be equal to the dimension of the domain.

06:14 Okay, now this guy is equal to four, as i discussed before.

06:17 Okay? this guy, which is just by definition the dimension of the kernel is equal to 1.

06:26 So the rank of t, which is the dimension of the image of t, is equal to 3.

06:33 The dimension of the image of t is equal to 3.

06:37 So it cannot be possibly m2r, right? because m2r, the code domain, is four dimensional.

06:46 Okay...

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