Question

Let \[ v_1 = \begin{pmatrix} 1 \\ -2 \\ 7 \end{pmatrix}, \quad v_2 = \begin{pmatrix} 2 \\ -2 \\ 3 \end{pmatrix} \] be two vectors in \(\mathbb{R}^3\). Then which of the following is true? 

• A. $\{v_1, v_2\}$ is a basis of $\mathbb{R}^3$
• B. $\{v_1, v_2\}$ is a basis of $\mathbb{R}^2$
• C. $\{v_1, v_2\}$ is neither linearly independent, nor it spans $\mathbb{R}^3$
• D. $\{v_1, v_2\}$ is a basis of the space span $\{v_1, v_2\}$

Hint:

Any set of $k$ linearly independent vectors in $\mathbb{R}^n$ (where $k \le n$) automatically forms a basis for the $k$-dimensional subspace that they span.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
A set of vectors $S$ forms a basis for a vector space $V$ if:
1. The vectors in $S$ are linearly independent.
2. The vectors in $S$ span the space $V$.

Step 2: Detailed Explanation:
Let us analyze the properties of the given vectors:
- First, check for linear independence:
Two vectors are linearly dependent if and only if one is a scalar multiple of the other ($v_2 = c \cdot v_1$).
If we compare the components:
\[ 2 = c \cdot 1 \implies c = 2 \]
But for the second component:
\[ -2 \neq 2 \cdot (-2) \]
Since no single scalar $c$ exists, $v_1$ and $v_2$ are linearly independent.
- Let us evaluate the options:
- (A) Basis of $\mathbb{R}^3$: Any basis of $\mathbb{R}^3$ must contain exactly 3 vectors. Since we only have 2 vectors, they cannot span $\mathbb{R}^3$ and thus cannot form its basis.
- (B) Basis of $\mathbb{R}^2$: The vectors are 3-dimensional, so they belong to $\mathbb{R}^3$, not $\mathbb{R}^2$.
- (C) "Neither linearly independent...": Since they are linearly independent, this statement is false.
- (D) "$\{v_1, v_2\}$ is a basis of the space span $\{v_1, v_2\}$": By definition, any linearly independent set of vectors serves as a basis for the subspace that it spans.
Thus, statement (D) is true.

Step 3: Final Answer
The correct option is (D).