For each of the following, decide whether the given transformation is linear. If yes, write a formal proof to show this. If not, give a specific counterexample to one of the properties of linearity.

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

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



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 \( T([1; 0]) + T([-1; 0]) \neq T([0; 0]) \).

---

### 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**.

---

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.

For each of the following, decide whether the given transformation is linear. If yes, write a formal proof to show this. If not, give a specific counterexample to one of the properties of linearity.
a) T([a;b]) = [|a|; b]
b) T([a;b]) = [a+b;a-b]

Explore whether the transformations T([a;b]) = [|a|; b] and T([a;b]) = [a+b;a-b] are linear. This solution covers the properties of linearity, including additivity and homogeneity, with formal proofs and counterexamples. Suitable for undergraduate students studying Linear Algebra.

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