Questions: Writing In Exercises 7-14, explain why S is not a basis for R^2. 7. S = (1,2),(1,0),(0,1)

Solution Steps

To determine if a set \( S \) is a basis for \( \mathbb{R}^2 \), we need to check two conditions: the set must be linearly independent and it must span \( \mathbb{R}^2 \). A set of vectors is linearly independent if no vector in the set can be written as a linear combination of the others. A set spans \( \mathbb{R}^2 \) if any vector in \( \mathbb{R}^2 \) can be expressed as a linear combination of the vectors in the set. Since \( \mathbb{R}^2 \) is a 2-dimensional space, a basis for \( \mathbb{R}^2 \) must consist of exactly 2 linearly independent vectors. The set \( S \) has 3 vectors, which means it cannot be a basis for \( \mathbb{R}^2 \).

The set of vectors is given as \( S = \{(1, 2), (1, 0), (0, 1)\} \). This set contains three vectors in \( \mathbb{R}^2 \).

To determine if \( S \) can be a basis for \( \mathbb{R}^2 \), we calculate the rank of the matrix formed by these vectors. The rank of the matrix is found to be \( \text{rank}(S) = 2 \).

A basis for \( \mathbb{R}^2 \) must consist of exactly 2 linearly independent vectors. Since the set \( S \) contains 3 vectors, it cannot be a basis for \( \mathbb{R}^2 \) despite having a rank of 2. Therefore, the condition for being a basis is not satisfied.

Final Answer