For each of the following pairs of linear functions S, T: ℝ^2 →ℝ^2, describe the compositions S

For each of the following pairs of linear functions S, T:...

For each of the following pairs of linear functions $S, T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, describe the compositions $S \circ T$ and $T \circ S$. Do the functions commute? (a) $S=$ counterclockwise rotation by $60^{\circ} ; T=$ clockwise rotation by $120^{\circ}$; (b) $S=$ reflection in the line $y=x ; T=$ rotation by $180^{\circ}$; (c) $S=$ reflection in the $x$-axis; $T=$ reflection in the $y$-axis; (d) $S=$ reflection in the line $y=x ; T=$ reflection in the line $y=2 x$; (e) $S=$ orthogonal projection on the $x$-axis; $T=$ orthogonal projection on the $y$-axis; (f) $S=$ orthogonal projection on the $x$-axis; $T=$ orthogonal projection on the line $y=x$; (g) $S=$ orthogonal projection on the $x$-axis; $T=$ rotation by $180^{\circ}$; (h) $S=$ orthogonal projection on the $x$-axis; $T=$ counterclockwise rotation by $90^{\circ}$; (i) $S=$ orthogonal projection on the line $y=-2 x ; T=$ reflection in the line $y=x$.

For each of the following pairs of linear functions $S, T: \mathbb{R}^2 \rightarrow \mathbb{R}^2$, describe the compositions $S \circ T$ and $T \circ S$. Do the functions commute?
(a) $S=$ counterclockwise rotation by $60^{\circ} ; T=$ clockwise rotation by $120^{\circ}$;
(b) $S=$ reflection in the line $y=x ; T=$ rotation by $180^{\circ}$;
(c) $S=$ reflection in the $x$-axis; $T=$ reflection in the $y$-axis;
(d) $S=$ reflection in the line $y=x ; T=$ reflection in the line $y=2 x$;
(e) $S=$ orthogonal projection on the $x$-axis; $T=$ orthogonal projection on the $y$-axis;
(f) $S=$ orthogonal projection on the $x$-axis; $T=$ orthogonal projection on the line $y=x$;
(g) $S=$ orthogonal projection on the $x$-axis; $T=$ rotation by $180^{\circ}$;
(h) $S=$ orthogonal projection on the $x$-axis; $T=$ counterclockwise rotation by $90^{\circ}$;
(i) $S=$ orthogonal projection on the line $y=-2 x ; T=$ reflection in the line $y=x$.

Key Concepts

Linear Transformations

Linear transformations are functions between vector spaces that preserve the operations of vector addition and scalar multiplication. In the context of R², these functions can be represented by 2×2 matrices, and they include rotations, reflections, and projections. Understanding the structure and properties of linear transformations is essential for analyzing how they affect the geometry of the plane.

Composition of Linear Transformations

The composition of linear transformations involves applying one transformation after another. Since each linear transformation can be represented by a matrix, the composition corresponds to matrix multiplication. Analyzing the composition gives insight into the cumulative effect of sequential transformations, and it is fundamental in understanding more complex operations in vector spaces.

Rotations in the Plane

Rotations are linear transformations that turn every vector in the plane by a certain angle about the origin. They preserve lengths and angles, making them isometries. The rotation angle, whether clockwise or counterclockwise, determines the specific matrix representation and the resulting orientation of the plane after the transformation.

Reflections in the Plane

Reflections are transformations that 'mirror' vectors across a specified line. In R², reflecting a point over a line results in a new point that is the same distance from the line as the original point but on the opposite side. Reflections change the orientation but preserve distances, and they are key examples of orthogonal transformations.

Orthogonal Projections

Orthogonal projections are linear transformations that map every point in the space onto a subspace (such as a line) in a manner that the difference between the original point and the projected point is perpendicular to the subspace. In R², common projections include those onto the x-axis or onto lines like y = x, and they are used to reduce dimensionality while preserving as much information as possible.

Commutativity of Linear Maps

The concept of commutativity in linear maps refers to the property that two transformations, S and T, satisfy S?T = T?S. When linear transformations commute, the order in which they are applied does not affect the final outcome of the transformation. Determining whether compositions commute is important for understanding the underlying algebraic structure and possible symmetries within the system.

Transcript

00:01 Okay, the key idea for this question is, if you're ever asked if two transformations commute, that is t1 composed with t2 is t2 composed with t1, to check that, all you have to do is check if the matrices, the standard matrices for the transformations commute.

00:21 So a times b is equal to b times a.

00:24 So again, transformations commute if and only if their standard matrices commute.

00:29 So if we're asking if t1 and t2 commute where t1 is the reflection about the x axes, sorry, a reflection about y equals x, and t2 is projection onto the x axes, the first thing we need to do is noor matrices for these reflections.

00:48 So t1 reflecting about the line y equals x, that has the standard matrix 0110, and then t2 projecting onto the x -axis has the standard matrix 1 -0 -0.

01:07 So now, again, this is a and this is b.

01:14 So now all we have to do is perform a matrix multiplication.

01:19 A -b is going to look like 0 -110 times by 1 -0 -0 -0.

01:30 And if we do that calculation, we're going to get 0 times 1 plus 1 times 0, which is 0.

01:40 Second column in the first row is going to be 0, 0 times 0 plus 1 times 0.

01:48 Down here in the bottom left, we're going to have 1 times 1 plus 0 times 0, which is 1.

01:56 And then the final entry we have 1 times 0 plus 0 times 0 which is 0.

02:04 So that's what a -b is.

02:07 And then we also need to calculate b times a in the reverse order.

02:14 And so we have 1 -0 -0 -0.

02:18 That's b times a, 0 -1 -1 -0.

02:23 And remember with matrix multiplication, it's not a given that these two things are actually equal.

02:28 We have to actually do the computation.

02:31 Okay, so if i multiply the matrices in this order, 1 times 0 plus 0 times 1 plus 0 times 0 gets me 1.

02:43 0 times 0 plus 0 times 1 gets me 0.

02:50 And finally, 0 times 1 plus 0 times 0 gets me 0.

02:56 So now these two matrices are not equal.

03:05 So that tells us t1, excuse me, t1 composed with t2 is not equal to t2 composed with t1.

03:20 So those transformations do not commute.

03:25 All righty, let's check another problem here.

03:29 In this case, t1 is reflection about the x axes.

03:33 Let's see if we can remember what that matrix looks like.

03:36 So reflecting about the x axes.

03:39 We're going to make the y coordinate 0, so that matrix is going to look like 1 -0 -0 -negative 1.

03:49 And then reflecting about the line y -equals x, that's the one we saw earlier...

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