Question

Let $W$ be a subspace of the vector space $\mathbb{R}^3$ over the field $\mathbb{R}$ spanned by 

Which one of the following vectors lies in $W$? 
 

• A.
• B.
• C.
• D.

Hint:

To check if a vector lies in a subspace, express it as a linear combination of the spanning vectors and verify if real coefficients exist.

Answer 1

The Correct Option is C

Step 1: Condition for a vector to lie in $W$.
A vector $\mathbf{v} = (x, y, z)^T$ lies in $W$ if there exist scalars $a$ and $b$ such that \[ \mathbf{v} = a \begin{pmatrix} 0 \\ -1 \\ 2 \end{pmatrix} + b \begin{pmatrix} 2 \\ -1 \\ 0 \end{pmatrix}. \]
Step 2: Write component-wise equations.
\[ \begin{cases} x = 2b, \\ y = -a - b, \\ z = 2a. \end{cases} \]
Step 3: Substitute from given vector $(1, -1, 1)$.
\[ 1 = 2b \Rightarrow b = \frac{1}{2}, \quad z = 2a = 1 \Rightarrow a = \frac{1}{2}. \] Now check $y = -a - b = -\frac{1}{2} - \frac{1}{2} = -1$, which matches the given $y$. Hence, $\begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix} \in W$.
Step 4: Conclusion.
The vector $\begin{pmatrix} 1 \\ -1 \\ 1 \end{pmatrix}$ lies in $W$.