1. Which of the following statements is TRUE?
a) Every matrix is row equivalent to a unique matrix in row echelon form.
b) The number of vectors in Span{u, v, w} is 3.
c) If a system of linear equations has no free variables, then it has a unique solution.
d) If the matrix equation AX = b is consistent, then b is in the span of the columns of A.

The correct statement is:

**d) If the matrix equation \( AX = b \) is consistent, then \( b \) is in the span of the columns of \( A \).**

### Explanation:
- **a) False**: Every matrix is **not** row equivalent to a **unique** matrix in row echelon form; it is row equivalent to a unique matrix in **reduced row echelon form (RREF)**, not row echelon form, because row echelon form is not unique.
  
- **b) False**: The number of vectors in **Span\{u, v, w\}** is not necessarily 3. The span of vectors represents all linear combinations of those vectors, which could span a lower-dimensional space, depending on whether the vectors are linearly dependent.

- **c) False**: A system of linear equations without free variables does not always imply a unique solution. It implies there are no free variables, which typically suggests a unique solution, but this isn't always the case if the system is inconsistent.

- **d) True**: If \( AX = b \) is consistent, it means that the equation has at least one solution. This implies that \( b \) is in the column space (or span) of the matrix \( A \)'s columns.

Would you like more details or have any questions about these explanations?

### Related Questions:
1. What conditions guarantee that a matrix has a unique row echelon form?
2. How can you determine the dimension of a span of vectors?
3. What are free variables, and how do they affect the solution set of a system?
4. What is the difference between row echelon form and reduced row echelon form?
5. How can you tell if a vector is in the span of a set of vectors?

**Tip:** When solving matrix equations, always check the consistency of the system by verifying if the vector \( b \) is in the column space of \( A \).

Which of the following statements is TRUE?
a) Every matrix is row equivalent to a unique matrix in row echelon form.
b) The number of vectors in Span{u, v, w} is 3.
c) If a system of linear equations has no free variables, then it has a unique solution.
d) If the matrix equation AX = b is consistent, then b is in the span of the columns of A.

Explore the correct statement about matrix equations, vector spans, and solutions to linear systems. Learn why the statement 'If the matrix equation AX = b is consistent, then b is in the span of the columns of A' is true, and understand the reasoning behind the falsehood of other statements in the context of linear algebra.

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