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# Image of a subset under a transformation

Exploring what happens to a subset of the domain under a transformation. Created by Sal Khan.

## Want to join the conversation?

• It should be + sign not -...please modify • What is a subset? •  First, you need to have a set for the term subset to mean anything. A set is a collection of elements. A subset of a set contains elements only from that set.

For example, set A = {1,2,3,4,5,6} and set B = {4,5}. B is a subset of A. A is not a subset of B.

My examples above are finite sets, but you can have infinite sets. For instance, R^2 is the set of all points in the plane. In this case, the set C = {[4,5],[6,7]} is a set with 2 points [4,5] and [6,7], and C is a subset of R^2.
• Why do you need to add X0 for the L0? Does not the X1 - X0 equals the L0? • At , why is it a minus sign after T(X0) instead of a + sign in the T(L0) equation, since the original equation of L0 has a + sign after X0 • In past videos, we would just draw the vector between x1 and x0 as (x1-x0), the difference between the two vectors. The orange arrow Lo was just (x1-x0). How come when we define the orange line this time, we have to add in a starting position x0 and then scale (x1-x0) by t? Fundamentally, I understand why starting at x0 and then adding a scale of (x1-x0) works, but what's the difference if we just said L0 = (x1-x0)? • I believe it's because you can do different kinds of things with these vectors than you can do with points and distances between points. A vector is very similar to a directed line segment (the difference between 2 points "P2 - P1").

A fixed vector seems to me to be a difference between two points, after you perform the subtraction, not before as in the equation for a plane n dot (v - v0), where the properties of both directed line segments and vectors are maintained.
(1 vote)
• at wouldn't it be accurate to say that L1 = t(x1) where -1 <= t <= 1 ?

Proof: t(x1) = x2 where t = -1 • Yes, it is correct to say that L1 = { t(x1) | -1 <= t <= 1 }
This is true because x2 = -x1

It should be noted, however, that you defined L1 from x2 to x1, while Sal defined it as x1 to x2. It doesn't make a difference, but I just wanted to inform you as to the direction of your L1. If you wanted to define L1 the exact same way, you could just flip t like this:

L1 = { -t(x1) | -1 <= t <= 1 } or L1 = { t(x2) | -1 <= t <= 1 }
That way when t = -1, L1 = -(-1(x1)) = x1, and when t=1, L1 = -(1(x1)) = -x1 = x2
• At why is it "X0-t..." and "not + t..."? • Sal seems to be saying that the sets L_0, L_1, and L_2 consist of spans that start at x_0, x_1, and x_2 and span the directed line segments connecting these points (In both the untransformed and transformed sets).

However, take one vector from these 3 sets, say "x0 + (3/4)(x1-x0)" = "(-2, -2) + (3/4)(0, 4)". The tail of (0, 4) is at the head of (-2, -2) by vector addition. So we have (0, 3) added to (-2, -2). But where is the tail of (-2, -2)? If it's not anywhere (which makes sense, because it's a vector), then we don't get lines in R^2 as drawn by Sal. If it's at the origin, then all the vectors in all 3 sets are from the origin and go different directions and sort of fill in the triangle defined by the (transformed or untransformed) points.

This isn't like the definition of a plane in R^3 where we took the difference between vectors, which were understood to have their tails at the origin, and got vectors that all originated from the same point "n dot (x - x0)". In this case we don't get vectors on the lines, rather, if we understand the vectors x0, x1, and x3 to start at the origin, then all the vectors in L1, L2, and L3 do also, no?

I think if e.g. "x0 + ..." were left out of the set L0, then L0 would actually be the line connecting x0 and x1; and similarly for L1 and L2. Does anyone disagree? • I do disagree. I will try to explain my thoughts. First, a terminology note: the span of a vector is all linear combinations of that vector (giving a line, not a line segment). It doesn't seem like that's what you are describing, so I won't use that term in this case.

If I understand your question, you want to know why L0 is the line segment between (-2, -2) and (-2, 2) rather than a different set definition, lets call K0, which would be K0 = {t(x1 – x0) : 0 ≤ t ≤ 1}. So let's consider K0.

Let t = 0. Then a vector in K0 must be (0, 0). Right away we have an issue, since the line going from the tip of x0 to the tip of x1 certainly does not go through the origin. K0 cannot describe our line.

To understand why L0 describes our line, start with x1 - x0. That gives us a vector of the correct length and slope (or non-slope in this case since it's a vertical line). But we want a set of all the vectors, so we scale by a constant Sal called t where 0 ≤ t ≤ 1. What this means is that when t = 1, we have a vector of the full possible length. When t = 0 we have a zero length vector, and as t ranges in-between, we get all lengths in-between. This is K0.

But we still need to get them in the right spot. So we take all those vectors in the set K0, and we add x0 to each. This moves them over - L0 is the line segment parallel to K0 which runs through the tip of x0. And that is what we wanted.

These types of sets are called affine sets. They are not generally subspaces, but they have a lot of other roles.  