For each of the following statements determine whether it is True (meaning it is true in every case)
or False (meaning there is at least one example where it is not true). If it is true, provide a short
justification for why it is true (one or two lines will suffice). If it is false, provide a counterexample
to the statement or a short explanation of why it’s false (it will always be possible to come up
with a small and simple counterexample).
(a) If two matrices A and B have the same reduced row echelon form, then the equations Ax = 0
and Bx = 0 must have the same solutions.
(b) If two matrices A and B have the same reduced row echelon form, then the equations Ax = y
and Bx = y must have the same solutions for any vector y.
(c) If u is a linear combination of vectors v and w and v is a linear combination of vectors p,q,
and r, then u must be a linear combination of p,q, r, and w.
(d) A linear system with fewer unknowns than equations must have infinitely many solutions or
none.

Let's analyze each statement to determine whether it is true or false, providing justifications or counterexamples as needed.

**(a) If two matrices \( A \) and \( B \) have the same reduced row echelon form, then the equations \( Ax = 0 \) and \( Bx = 0 \) must have the same solutions.**

- **True.** The solution set to the equation \( Ax = 0 \) depends on the row space of \( A \). If two matrices have the same reduced row echelon form, they represent the same row space, which means the null spaces of \( A \) and \( B \) (i.e., the solution sets to \( Ax = 0 \) and \( Bx = 0 \)) are identical.

**(b) If two matrices \( A \) and \( B \) have the same reduced row echelon form, then the equations \( Ax = y \) and \( Bx = y \) must have the same solutions for any vector \( y \).**

- **True.** If \( A \) and \( B \) have the same reduced row echelon form, they represent the same linear transformation. Therefore, for any vector \( y \), the solution sets of \( Ax = y \) and \( Bx = y \) will be the same because both transformations apply to the same space.

**(c) If \( u \) is a linear combination of vectors \( v \) and \( w \) and \( v \) is a linear combination of vectors \( p, q, \) and \( r \), then \( u \) must be a linear combination of \( p, q, r, \) and \( w \).**

- **True.** Since \( u = a_1v + a_2w \) and \( v = b_1p + b_2q + b_3r \), by substitution, \( u \) can be expressed as:
  \[
  u = a_1(b_1p + b_2q + b_3r) + a_2w = (a_1b_1)p + (a_1b_2)q + (a_1b_3)r + a_2w,
  \]
  which is a linear combination of \( p, q, r, \) and \( w \).

**(d) A linear system with fewer unknowns than equations must have infinitely many solutions or none.**

- **False.** Consider the system:
  \[
  \begin{aligned}
  x_1 + x_2 &= 1 \\
  x_1 - x_2 &= 0
  \end{aligned}
  \]
  This system has more equations than unknowns (2 equations, 2 unknowns), but it has a unique solution: \( x_1 = 0.5, x_2 = 0.5 \). Therefore, a linear system with fewer unknowns than equations can have a unique solution, which refutes the statement.

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Would you like further details or have any questions?

Here are 5 related questions:
1. What conditions guarantee that a system of linear equations has a unique solution?
2. How does the rank of a matrix relate to the number of solutions in a linear system?
3. Can two different matrices with the same rank have different reduced row echelon forms?
4. How can you determine the solution space of a homogeneous linear system?
5. Why does a matrix having more rows than columns not always imply a system has no solutions?

**Tip:** Always check the rank of a matrix to determine the nature of the solutions to a system of linear equations.

Explore the concepts of matrix equivalence and linear combinations in linear algebra. Learn about reduced row echelon form, its properties, and how it affects the solutions of linear systems. Suitable for college-level students and beyond.

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