Determine whether the operators T1 and T2 commute; that is,...

Determine whether the operators $T_{1}$ and $T_{2}$ commute; that is, whether $T_{1} \circ T_{2}=T_{2} \circ T_{1}$. (a) $T_{1}: R^{2} \rightarrow R^{2}$ is the orthogonal projection onto the $x$ -axis, and $T_{2}: R^{2} \rightarrow R^{2}$ is the orthogonal projection onto the $y$ -axis. (b) $T_{1}: R^{2} \rightarrow R^{2}$ is the rotation about the origin through an angle of $\pi / 4,$ and $T_{2}: R^{2} \rightarrow R^{2}$ is the reflection about the $y$ -axis.

Determine whether the operators $T_{1}$ and $T_{2}$ commute; that is, whether $T_{1} \circ T_{2}=T_{2} \circ T_{1}$.
(a) $T_{1}: R^{2} \rightarrow R^{2}$ is the orthogonal projection onto the $x$ -axis, and $T_{2}: R^{2} \rightarrow R^{2}$ is the orthogonal projection onto the $y$ -axis.
(b) $T_{1}: R^{2} \rightarrow R^{2}$ is the rotation about the origin through an angle of $\pi / 4,$ and $T_{2}: R^{2} \rightarrow R^{2}$ is the reflection about the $y$ -axis.

Key Concepts

Reflection

A reflection is a type of linear transformation that mirrors points across a specified subspace, such as a line or a plane. Reflections are fundamental in the study of symmetries and are often used alongside rotations to analyze geometric and algebraic structures.

Orthogonal Projection

An orthogonal projection is a linear transformation that maps vectors onto a subspace in such a way that the difference between the vector and its projection is orthogonal to that subspace. This technique simplifies problems by reducing dimensions while preserving the structure of the vector space.

Rotation

A rotation is a linear transformation that turns every vector in a space around a fixed point by a certain angle. Rotations preserve lengths and angles, and are key tools in understanding symmetry and invariance in geometric contexts.

Operator Composition

Operator composition is the process of applying one linear operator after another. It involves performing the transformation defined by the first operator and then applying the second operator to the result. This concept is essential when exploring how different transformations interact.

Linear Operators

Linear operators are functions between vector spaces that preserve the operations of vector addition and scalar multiplication. They serve as the building blocks in linear algebra and are used to model many geometric and algebraic transformations.

Commutativity

Commutativity in the context of linear operators means that the order in which the operators are applied does not affect the final outcome. Two operators commute if their composition is the same regardless of the order, which is a central concept when studying the interplay between different transformations.

Transcript

00:01 Okay, the key idea, if you're asked if two transformations commute, is that that is the same as t1 composed with t2 is t2 composed with t1.

00:13 And to check that, if you have the standard matrices for the transformations, you just need to do some matrix multiplication and check does a -b -equal b -a.

00:26 So for this first problem here, if t -1 is the projection onto the x -axis, and t2 is the projection onto the y axes, and we're asked, do they commute? what we need to do is figure out what their matrices are first, and then check if their matrices commute.

00:48 So if we remember projecting onto the x -axis, that has a matrix that looks like 1 -0 -0.

00:57 So 1 -0 -0.

01:00 And projecting onto the y axes has the matrix 0 -0 -0 -1.

01:10 I want to make the x -coordinate 0.

01:16 All right.

01:17 So what we need to do now is just check what is a -b and what is b -a and are those equal.

01:27 So let's just write down what is a times b.

01:30 So 1 -0 -0 -0 times by 0 -0 -1.

01:38 And if we do that calculation, we're not going to notice we're going to get 0 -0 -0.

01:46 All right.

01:51 So again, just doing some matrix multiplication there.

01:54 And for b times a, remember, it's not always the case that matrices commute.

02:00 So we actually have to do the calculation, 0 -0 -0 -1 times 1.

02:06 0 -0.

02:09 Again, do that matrix multiplication.

02:12 0 times 1, 1 times 0 gets me 0, 0 times 1 times 1 is going to get me 0 here.

02:22 And then also the last one is 0.

02:26 All righty, so these two matrices are equal.

02:31 And so that means that t1 and t2 commute.

02:46 So again, i just need to check.

02:48 These two matrices are equal.

02:51 They are both equal to the zero matrix.

02:53 So t1 and t2 commute.

02:56 Great.

02:57 Let's look at another example here.

02:59 Again, the key idea is i need to get their standard matrices and check if the standard matrices commute.

03:09 So for the second part of the problem, i'm going to rotate through an angle of pi over four.

03:15 So again, let's get the standard matrix for that...

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