Which of the following subsets of R2 are spanning sets? Explain your answer.(a) {(−1, 0), (0, 1)}, (b) {(−1, 1), (1, −1)}, (c) {(−1, 0), (1, 0), (−1, 1)
Question
Which of the following subsets of R² are spanning sets? Explain your answer.
(a) {(-1, 0), (0, 1)}
(b) {(-1, 1), (1, -1)}
(c) {(-1, 0), (1, 0), (-1, 1)}
Solution
(a) {(−1, 0), (0, 1)}: This is a spanning set. In R2, any set of two vectors that are not scalar multiples of each other can span the space. Here, the two vectors are linearly independent (not multiples of each other), so they can generate any other vector in R2 through linear combinations.
(b) {(−1, 1), (1, −1)}: This is also a spanning set. Although these vectors are negatives of each other, they are still linearly independent. They can generate any other vector in R2 through linear combinations.
(c) {(−1, 0), (1, 0), (−1, 1)}: This is also a spanning set. Even though the first two vectors are scalar multiples of each other and therefore linearly dependent, the third vector is not a multiple of the first two. Therefore, these vectors can still generate any other vector in R2 through linear combinations.
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