Question

Let \( u, v \in \mathbb{R}^4 \) be such that \( u = \begin{pmatrix} 1 \\ 2 \\ 3 \\ 5 \end{pmatrix} \) and \( v = \begin{pmatrix} 5 \\ 3 \\ 2 \\ 1 \end{pmatrix} \). Then the equation \( u^T x = v \) has

• A. infinitely many solutions
• B. no solution
• C. exactly one solution
• D. exactly two solutions

Hint:

When solving linear equations, make sure the dimensions of both sides are compatible. In this case, a scalar cannot equal a vector, indicating no solution.

Answer 1

The Correct Option is B

Step 1: Understanding the system.
The given equation is \( u^T x = v \), where \( u^T \) is a 1x4 row vector and \( v \) is a 4x1 column vector. The vector \( u^T \) represents a linear combination of the components of the vector \( x \).
Step 2: Analyzing consistency.
We need to check whether this system is consistent. To do this, we observe that the dimensions of the equation imply that \( u^T x \) results in a scalar, while \( v \) is a vector. The equation cannot be satisfied since a scalar (from \( u^T x \)) cannot equal a 4-dimensional vector. Hence, the system is inconsistent.
Step 3: Conclusion.
Since no solution exists for this system, the correct answer is (B).