QUESTION 12.1 Basis for the 2D Space Choose one $\cdot 4$ points Which set $S=\left\{\overrightarrow{v_{1}}, \overrightarrow{v_{2}}\right\}$ form a basis for $\mathbb{R}^{2}$ ? $\overrightarrow{v_{1}}=(2,1), \overrightarrow{v_{2}}=(0,0)$ $\overrightarrow{v_{1}}=(2,0), \overrightarrow{v_{2}}=(0,1)$ $\overrightarrow{v_{1}}=(1,2), \overrightarrow{v_{2}}=(4,8)$
Solution
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 ：\(\boxed{\text{Final Answer: The set }\{\overrightarrow{v_{1}}, \overrightarrow{v_{2}}\}\text{ with }\overrightarrow{v_{1}}=(2,0)\text{ and }\overrightarrow{v_{2}}=(0,1)\text{ forms a basis for }\mathbb{R}^{2}.}\)
