Question:
Let \( u, v \in \mathbb{R}^4 \) be such that
\[
u = \begin{pmatrix}
1 \\ 2 \\ 3 \\ 5
\end{pmatrix}
\quad \text{and} \quad
v = \begin{pmatrix}
5 \\ 3 \\ 2 \\ 1
\end{pmatrix}
\]
Then the equation \( u^T x = v \) has
Let \( u, v \in \mathbb{R}^4 \) be such that
\[ u = \begin{pmatrix} 1 \\ 2 \\ 3 \\ 5 \end{pmatrix} \quad \text{and} \quad v = \begin{pmatrix} 5 \\ 3 \\ 2 \\ 1 \end{pmatrix} \]
Then the equation \( u^T x = v \) hasShow 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.
Updated On: Dec 15, 2025
- infinitely many solutions
- no solution
- exactly one solution
- exactly two solutions
Show Solution
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The Correct Option is B
Solution and Explanation
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).
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).
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