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: 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.