Suppose 𝑣1
, … , 𝑣𝑛
is a basis of 𝑉 and 𝑤1
, … , 𝑤𝑚 is a basis of 𝑊.
(a) Show that if 𝑆, 𝑇 ∈ ℒ(𝑉, 𝑊), then ℳ(𝑆 + 𝑇) = ℳ(𝑆) + ℳ(𝑇).
(b) Show that if 𝜆 ∈ 𝐅 and 𝑇 ∈ ℒ(𝑉, 𝑊), then ℳ(𝜆𝑇) = 𝜆ℳ(𝑇)

We are given two vector spaces \( V \) and \( W \) with bases \( v_1, \dots, v_n \) for \( V \) and \( w_1, \dots, w_m \) for \( W \). Let \( S, T \in \mathcal{L}(V, W) \), meaning that \( S \) and \( T \) are linear maps from \( V \) to \( W \). We are tasked with proving two properties of the matrix representations \( \mathcal{M}(S) \) and \( \mathcal{M}(T) \).

### Part (a): Show that \( \mathcal{M}(S + T) = \mathcal{M}(S) + \mathcal{M}(T) \).

#### Proof:

Let \( S, T: V \to W \) be linear maps. We need to show that the matrix of the sum \( S + T \) is the sum of the matrices of \( S \) and \( T \). 

The matrix of a linear map \( S \in \mathcal{L}(V, W) \) relative to the bases \( \{v_1, \dots, v_n\} \) of \( V \) and \( \{w_1, \dots, w_m\} \) of \( W \) is constructed as follows: the \( (i, j) \)-th entry of the matrix \( \mathcal{M}(S) \) is the coefficient of \( w_i \) in the expansion of \( S(v_j) \) in terms of the basis \( \{w_1, \dots, w_m\} \).

Similarly, the matrix for \( T \) is defined in the same way. For each \( v_j \in V \),
\[
S(v_j) = \sum_{i=1}^{m} s_{ij} w_i \quad \text{and} \quad T(v_j) = \sum_{i=1}^{m} t_{ij} w_i,
\]
where \( s_{ij} \) and \( t_{ij} \) are the coefficients in the basis \( \{w_1, \dots, w_m\} \).

Now consider \( (S + T)(v_j) \):
\[
(S + T)(v_j) = S(v_j) + T(v_j) = \sum_{i=1}^{m} s_{ij} w_i + \sum_{i=1}^{m} t_{ij} w_i = \sum_{i=1}^{m} (s_{ij} + t_{ij}) w_i.
\]
Thus, the \( (i, j) \)-th entry of the matrix \( \mathcal{M}(S + T) \) is \( s_{ij} + t_{ij} \), which is precisely the sum of the \( (i, j) \)-th entries of \( \mathcal{M}(S) \) and \( \mathcal{M}(T) \).

Therefore, we have:
\[
\mathcal{M}(S + T) = \mathcal{M}(S) + \mathcal{M}(T).
\]

### Part (b): Show that \( \mathcal{M}(\lambda T) = \lambda \mathcal{M}(T) \) for \( \lambda \in \mathbb{F} \).

#### Proof:

Let \( T: V \to W \) be a linear map, and \( \lambda \in \mathbb{F} \) be a scalar. We need to show that the matrix of \( \lambda T \) is \( \lambda \) times the matrix of \( T \).

For each basis vector \( v_j \in V \), the action of \( T \) on \( v_j \) is given by:
\[
T(v_j) = \sum_{i=1}^{m} t_{ij} w_i,
\]
where \( t_{ij} \) are the entries of the matrix \( \mathcal{M}(T) \).

Now consider \( (\lambda T)(v_j) \):
\[
(\lambda T)(v_j) = \lambda T(v_j) = \lambda \sum_{i=1}^{m} t_{ij} w_i = \sum_{i=1}^{m} (\lambda t_{ij}) w_i.
\]
Thus, the \( (i, j) \)-th entry of the matrix \( \mathcal{M}(\lambda T) \) is \( \lambda t_{ij} \), which is precisely \( \lambda \) times the \( (i, j) \)-th entry of \( \mathcal{M}(T) \).

Therefore, we have:
\[
\mathcal{M}(\lambda T) = \lambda \mathcal{M}(T).
\]

---

These two results are standard properties of matrix representations of linear maps. In summary:
- The matrix of the sum of two linear maps is the sum of their matrices.
- The matrix of the scalar multiple of a linear map is the scalar multiple of its matrix.

Let me know if you'd like further clarification or details!

### Related Questions:
1. How are matrix representations related to changes in basis for vector spaces?
2. How do the properties of matrix addition and scalar multiplication extend to higher-dimensional spaces?
3. Can these results be generalized for infinite-dimensional vector spaces?
4. How do linear transformations preserve vector space structure?
5. How is the rank of a matrix affected by linear maps and their combinations?

### Tip:
When dealing with matrices, remember that the matrix representation of a linear map depends on the choice of basis. Changing the basis changes the matrix, though the map remains the same.

Suppose 𝑣1, … , 𝑣𝑛 is a basis of 𝑉 and 𝑤1, … , 𝑤𝑚 is a basis of 𝑊. (a) Show that if 𝑆, 𝑇 ∈ ℒ(𝑉, 𝑊), then ℳ(𝑆 + 𝑇) = ℳ(𝑆) + ℳ(𝑇). (b) Show that if 𝜆 ∈ 𝐅 and 𝑇 ∈ ℒ(𝑉, 𝑊), then ℳ(𝜆𝑇) = 𝜆ℳ(𝑇)

Explore how matrix representations of linear maps in vector spaces follow specific properties. Learn how to show that ℳ(S + T) = ℳ(S) + ℳ(T) and ℳ(λT) = λℳ(T) with detailed examples and proofs. Ideal for undergraduate students studying linear algebra.

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