To determine whether the given transformations are linear, we need to check if they satisfy the following two properties of linearity:

1. Additivity: $T(\mathbf{v}_1 + \mathbf{v}_2) = T(\mathbf{v}_1) + T(\mathbf{v}_2)$ for any vectors $\mathbf{v}_1$ and $\mathbf{v}_2$.
2. Homogeneity: $T(c\mathbf{v}) = cT(\mathbf{v})$ for any scalar $c$ and vector $\mathbf{v}$.

Part (a) $T([a;b]) = [|a|; b]$

We need to verify if this transformation is linear by checking the properties.

1. Additivity:

Let $\mathbf{v}_1 = [a_1; b_1]$ and $\mathbf{v}_2 = [a_2; b_2]$. Then,

$\mathbf{v}_1 + \mathbf{v}_2 = [a_1 + a_2; b_1 + b_2]$ [ T(\mathbf{v}_1 + \mathbf{v}_2) = T([a_1 + a_2; b_1 + b_2]) = [|a_1 + a_2|; b_1 + b_2] ] On the other hand, compute $T(\mathbf{v}_1) + T(\mathbf{v}_2)$: [ T(\mathbf{v}_1) = [|a_1|; b_1] \quad \text{and} \quad T(\mathbf{v}_2) = [|a_2|; b_2] ] $T(\mathbf{v}_1) + T(\mathbf{v}_2) = [|a_1| + |a_2|; b_1 + b_2]$

Clearly, $[|a_1 + a_2|; b_1 + b_2] \neq [|a_1| + |a_2|; b_1 + b_2]$ in general. For example, if $a_1 = 1$ and $a_2 = -1$, then $|a_1 + a_2| = 0$, but $|a_1| + |a_2| = 2$.

Thus, additivity fails, so $T$ is not linear.

Conclusion: $T([a;b]) = [|a|; b]$ is not linear. A counterexample is $T([1; 0]) + T([-1; 0]) \neq T([0; 0])$.

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Part (b) $T([a;b]) = [a + b; a - b]$

Now, let's check the properties for this transformation.

1. Additivity:

Let $\mathbf{v}_1 = [a_1; b_1]$ and $\mathbf{v}_2 = [a_2; b_2]$. Then,

$\mathbf{v}_1 + \mathbf{v}_2 = [a_1 + a_2; b_1 + b_2]$ [ T(\mathbf{v}_1 + \mathbf{v}_2) = T([a_1 + a_2; b_1 + b_2]) = [(a_1 + a_2) + (b_1 + b_2); (a_1 + a_2) - (b_1 + b_2)] = [a_1 + b_1 + a_2 + b_2; a_1 - b_1 + a_2 - b_2] ] Now, compute $T(\mathbf{v}_1) + T(\mathbf{v}_2)$: [ T(\mathbf{v}_1) = [a_1 + b_1; a_1 - b_1] \quad \text{and} \quad T(\mathbf{v}_2) = [a_2 + b_2; a_2 - b_2] ] $T(\mathbf{v}_1) + T(\mathbf{v}_2) = [a_1 + b_1 + a_2 + b_2; a_1 - b_1 + a_2 - b_2]$

Since $T(\mathbf{v}_1 + \mathbf{v}_2) = T(\mathbf{v}_1) + T(\mathbf{v}_2)$, additivity holds.

2. Homogeneity:

Let $c$ be a scalar, and $\mathbf{v} = [a; b]$. Then,

$c\mathbf{v} = [ca; cb]$ [ T(c\mathbf{v}) = T([ca; cb]) = [ca + cb; ca - cb] = c[a + b; a - b] ] On the other hand, [ cT(\mathbf{v}) = c[a + b; a - b] = [c(a + b); c(a - b)] ]

Since $T(c\mathbf{v}) = cT(\mathbf{v})$, homogeneity holds.

Conclusion: $T([a;b]) = [a + b; a - b]$ is linear.

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Would you like more details or have any questions?

Here are 5 related questions:

1. Can a transformation be linear if it fails either additivity or homogeneity?
2. What are some examples of real-world linear transformations?
3. How can matrix representation help in understanding linear transformations?
4. What other properties can we check to verify linearity of a transformation?
5. How do non-linear transformations differ from linear ones in graphical representations?

Tip: A linear transformation can always be represented as a matrix multiplication, provided the transformation is from one vector space to another.