For Unit 1, Question 1, you are asked to prove that the following functions are linear transformations:

i) $T: \mathbb{R}^3 \to \mathbb{R}^3$ where $T(x, y, z) = (x - y + z, y - z)$.

ii) $T: \mathbb{R}^3 \to \mathbb{R}^3$ where $T(x_1, x_2, x_3) = (x_1 - x_2 + x_3, x_1 - x_2, x_1 - x_3)$.

A function $T: V \to W$ is a linear transformation if it satisfies the following two conditions for all vectors $\mathbf{v}, \mathbf{w} \in V$ and scalars $c \in \mathbb{R}$:

1. Additivity: $T(\mathbf{v} + \mathbf{w}) = T(\mathbf{v}) + T(\mathbf{w})$
2. Homogeneity (or scalar multiplication): $T(c \mathbf{v}) = c T(\mathbf{v})$

Let’s check each part:

Part i)

$T(x, y, z) = (x - y + z, y - z)$

1. Additivity: Let $T((x_1, y_1, z_1)) = (x_1 - y_1 + z_1, y_1 - z_1)$ and $T((x_2, y_2, z_2)) = (x_2 - y_2 + z_2, y_2 - z_2)$. Then: $T((x_1 + x_2, y_1 + y_2, z_1 + z_2)) = ( (x_1 + x_2) - (y_1 + y_2) + (z_1 + z_2), (y_1 + y_2) - (z_1 + z_2) )$ This simplifies to: $( (x_1 - y_1 + z_1) + (x_2 - y_2 + z_2), (y_1 - z_1) + (y_2 - z_2) )$ Which equals: $T(x_1, y_1, z_1) + T(x_2, y_2, z_2)$ Hence, additivity holds.

2. Homogeneity: Let $T(x, y, z) = (x - y + z, y - z)$. For scalar $c$: $T(c(x, y, z)) = T((cx, cy, cz)) = (cx - cy + cz, cy - cz)$ This simplifies to: $c(x - y + z, y - z) = c T(x, y, z)$ Hence, homogeneity holds.

Since both conditions are satisfied, part i) is a linear transformation.

Part ii)

$T(x_1, x_2, x_3) = (x_1 - x_2 + x_3, x_1 - x_2, x_1 - x_3)$

1. Additivity: Let $T((x_1, x_2, x_3)) = (x_1 - x_2 + x_3, x_1 - x_2, x_1 - x_3)$ and $T((y_1, y_2, y_3)) = (y_1 - y_2 + y_3, y_1 - y_2, y_1 - y_3)$. Then: $T((x_1 + y_1, x_2 + y_2, x_3 + y_3)) = ((x_1 + y_1) - (x_2 + y_2) + (x_3 + y_3), (x_1 + y_1) - (x_2 + y_2), (x_1 + y_1) - (x_3 + y_3))$ This simplifies to: $(x_1 - x_2 + x_3 + y_1 - y_2 + y_3, x_1 - x_2 + y_1 - y_2, x_1 - x_3 + y_1 - y_3)$ Which equals: $T(x_1, x_2, x_3) + T(y_1, y_2, y_3)$ Hence, additivity holds.

2. Homogeneity: Let $T(x_1, x_2, x_3) = (x_1 - x_2 + x_3, x_1 - x_2, x_1 - x_3)$. For scalar $c$: $T(c(x_1, x_2, x_3)) = T((cx_1, cx_2, cx_3)) = (cx_1 - cx_2 + cx_3, cx_1 - cx_2, cx_1 - cx_3)$ This simplifies to: $c(x_1 - x_2 + x_3, x_1 - x_2, x_1 - x_3) = c T(x_1, x_2, x_3)$ Hence, homogeneity holds.

Since both conditions are satisfied, part ii) is also a linear transformation.

Conclusion:

Both parts i) and ii) are linear transformations.

Would you like further details on this or have any specific questions? Here are a few related questions to explore:

1. How do you prove that a given function is not a linear transformation?
2. What is the significance of the matrix representation of a linear transformation?
3. Can a linear transformation be non-continuous?
4. How do you determine the kernel and image of a linear transformation?
5. What is the difference between a linear transformation and an affine transformation?

Tip: Understanding the properties of linear transformations (additivity and homogeneity) helps in easily proving whether a function is linear.