Question

question 2
which of the following is always true?
(a) if span{\vec{v}_1, \dots, \vec{v}_n} = \mathbb{r}^m, then n = m.
(b) if span{\vec{v}_1, \dots, \vec{v}_n} = \mathbb{r}^m, then n \geq m.
(c) there are vectors \vec{u}, \vec{v} \in \mathbb{r}^2 where \vec{u} = 2\vec{v} and {\vec{u}, \vec{v}} is a basis for \mathbb{r}^2.
(d) every set of 3 vectors spans \mathbb{r}^3.
(e) none of the above statements is always true.

Explanation:

Step1: Analyze Option A

The span of \( n \) vectors in \( \mathbb{R}^m \) being \( \mathbb{R}^m \) doesn't require \( n = m \). For example, in \( \mathbb{R}^2 \), we can have 3 vectors (e.g., \(

$$\begin{bmatrix}1\\0\end{bmatrix}$$

,

$$\begin{bmatrix}0\\1\end{bmatrix}$$

,

$$\begin{bmatrix}1\\1\end{bmatrix}$$

\)) whose span is \( \mathbb{R}^2 \), but \( n = 3>m = 2 \). So A is false.

Step2: Analyze Option B

To span \( \mathbb{R}^m \), we need at least \( m \) linearly independent vectors (by the basis theorem, a basis for \( \mathbb{R}^m \) has \( m \) vectors, and the span of a set containing a basis will be \( \mathbb{R}^m \)). If \( \text{span}\{\vec{v}_1,\dots,\vec{v}_n\}=\mathbb{R}^m \), then the set must contain at least \( m \) linearly independent vectors (otherwise, the dimension of the span would be less than \( m \)). So \( n\geq m \) (since we can have more than \( m \) vectors, as in the example for A). So B seems true, but let's check other options.

Step3: Analyze Option C

If \( \vec{u}=2\vec{v} \), then \( \vec{u} - 2\vec{v}=\vec{0} \), so \( \vec{u} \) and \( \vec{v} \) are linearly dependent. A basis for \( \mathbb{R}^2 \) must consist of 2 linearly independent vectors. So \( \{\vec{u},\vec{v}\} \) can't be a basis, so C is false.

Step4: Analyze Option D

A set of 3 vectors in \( \mathbb{R}^3 \) may be linearly dependent (e.g., \(

$$\begin{bmatrix}1\\0\\0\end{bmatrix}$$

,

$$\begin{bmatrix}2\\0\\0\end{bmatrix}$$

,

$$\begin{bmatrix}3\\0\\0\end{bmatrix}$$

\)), and their span is a line (1 - dimensional), not \( \mathbb{R}^3 \). So D is false.

Answer:

B. If \( \text{span}\{\vec{v}_1,\dots,\vec{v}_n\}=\mathbb{R}^m \), then \( n \geq m \)