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The problem is asking to prove a relationship involving the matrix \( A \) and a vector \( y \). Here's a breakdown of the problem:

### Given:
- A vector \( y = [y_1, y_2, \ldots, y_m] \) (assumed to be \( 1 \times m \)).
- A matrix \( A \) of dimension \( m \times n \) where \( A = [a_{ij}] \) with \( i = 1, 2, \ldots, m \) and \( j = 1, 2, \ldots, n \).
- \( P_i \) denotes the \( i \)-th row of \( A \).

### Objective:
Prove that:
\[
yA = \sum_{i=1}^{m} y_i P_i
\]

### Solution:
Let's analyze this equation step by step.

1. **Vector-Matrix Multiplication**: 
   - The multiplication of a row vector \( y = [y_1, y_2, \ldots, y_m] \) by the matrix \( A \) (which is \( m \times n \)) results in a row vector of size \( 1 \times n \).
   - Mathematically, the product \( yA \) is given by:
     \[
     yA = \left[ \sum_{i=1}^{m} y_i a_{i1}, \sum_{i=1}^{m} y_i a_{i2}, \ldots, \sum_{i=1}^{m} y_i a_{in} \right]
     \]

2. **Expressing in Terms of Rows**:
   - Each \( P_i \) is a row vector corresponding to the \( i \)-th row of matrix \( A \). Thus, \( P_i = [a_{i1}, a_{i2}, \ldots, a_{in}] \).
   - Therefore, \( y_i P_i \) is a row vector obtained by multiplying the scalar \( y_i \) with the row \( P_i \):
     \[
     y_i P_i = [y_i a_{i1}, y_i a_{i2}, \ldots, y_i a_{in}]
     \]
   - Summing these vectors for \( i \) from 1 to \( m \) gives:
     \[
     \sum_{i=1}^{m} y_i P_i = \left[ \sum_{i=1}^{m} y_i a_{i1}, \sum_{i=1}^{m} y_i a_{i2}, \ldots, \sum_{i=1}^{m} y_i a_{in} \right]
     \]

3. **Conclusion**:
   - Comparing the result of the vector-matrix multiplication \( yA \) and the sum \( \sum_{i=1}^{m} y_i P_i \), we observe that they are identical.
   - Thus, the identity is proven:
     \[
     yA = \sum_{i=1}^{m} y_i P_i
     \]

Would you like more details on any specific part of this proof, or do you have any questions?

### Follow-up Questions:
1. How would you describe the role of the row vector \( y \) in the multiplication with matrix \( A \)?
2. Can this proof be extended to any non-square matrices? Why or why not?
3. What happens if \( y \) is a column vector instead of a row vector?
4. How would the proof change if the matrix \( A \) was transposed?
5. In what situations could this identity be practically useful?

### Tip:
When multiplying matrices and vectors, keeping track of dimensions is key to ensuring the operations are valid and correctly interpreting the results.

Prove that yA = sum from i=1 to m of y_i P_i, where y is a vector and A is a matrix.

Explore the proof that yA equals the sum of y_i multiplied by each row vector P_i of matrix A. This problem is a fundamental concept in linear algebra, focusing on vector-matrix multiplication. Suitable for university-level students studying linear algebra.

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