QUESTION 12.1
Basis for the 2D Space Choose one $\cdot 4$ points
Which set $S=\{\overrightarrow{v_1},\overrightarrow{v_2}\}$ form a basis for ${\mathbb{R}}^2$ ?
$\overrightarrow{v_1}=(2,1),\overrightarrow{v_2}=(0,0)$
$\overrightarrow{v_1}=(2,0),\overrightarrow{v_2}=(0,1)$
$\overrightarrow{v_1}=(1,2),\overrightarrow{v_2}=(4,8)$

Step 1 ：Given the set $S=\{\overrightarrow{v_1},\overrightarrow{v_2}\}$, we are to determine which pair of vectors form a basis for ${\mathbb{R}}^2$.

Step 2 ：A basis for a vector space is a set of vectors that are linearly independent and that span the vector space. This means that no vector in the set can be written as a linear combination of the others, and any vector in the vector space can be written as a linear combination of the vectors in the set.

Step 3 ：Checking each pair of vectors:

Step 4 ：For $\overrightarrow{v_1}=(2,1),\overrightarrow{v_2}=(0,0)$: The second vector is the zero vector, which is always linearly dependent with any other vector. So this set cannot form a basis.

Step 5 ：For $\overrightarrow{v_1}=(2,0),\overrightarrow{v_2}=(0,1)$: These vectors are linearly independent (none of them can be written as a scalar multiple of the other), and they span ${\mathbb{R}}^2$ (any vector in ${\mathbb{R}}^2$ can be written as a linear combination of these two vectors). So this set can form a basis.

Step 6 ：For $\overrightarrow{v_1}=(1,2),\overrightarrow{v_2}=(4,8)$: These vectors are linearly dependent (the second can be written as a scalar multiple of the first), so this set cannot form a basis.

Step 7 ：Thus, the set $\{\overrightarrow{v_1},\overrightarrow{v_2}\}$ with $\overrightarrow{v_1}=(2,0)$ and $\overrightarrow{v_2}=(0,1)$ forms a basis for ${\mathbb{R}}^2$.

Step 8 ：$\boxed{{\displaystyle \text{Final Answer: The set }\{\overrightarrow{v_1},\overrightarrow{v_2}\}\text{ with }\overrightarrow{v_1}=(2,0)\text{ and }\overrightarrow{v_2}=(0,1)\text{ forms a basis for }{\mathbb{R}}^2.}}$