Which of the following subsets of R3 are subspaces?(a) The plane of vectors (b1, b2, b3) with b1 = b2.(b) The plane of vectors with b1 = 1.(c) The vectors with b1b2b3 = 0.
Question
Which of the following subsets of R3 are subspaces?(a) The plane of vectors (b1, b2, b3) with b1 = b2.(b) The plane of vectors with b1 = 1.(c) The vectors with b1b2b3 = 0.
Solution
To determine whether a subset of R3 is a subspace, it must satisfy three conditions:
- The zero vector is in the subset.
- The subset is closed under vector addition.
- The subset is closed under scalar multiplication.
Let's examine each subset:
(a) The plane of vectors (b1, b2, b3) with b1 = b2.
This subset includes the zero vector (0,0,0). If we add two vectors from this subset, we get another vector in the subset because the sum of the first and second components will still be equal. If we multiply a vector from this subset by a scalar, we still get a vector in the subset because the first and second components will still be equal. Therefore, this subset is a subspace of R3.
(b) The plane of vectors with b1 = 1.
This subset does not include the zero vector because the first component is always 1, not 0. Therefore, this subset is not a subspace of R3.
(c) The vectors with b1b2b3 = 0.
This subset includes the zero vector. However, it is not closed under vector addition. For example, the vectors (1,1,0) and (1,0,1) are in the subset, but their sum (2,1,1) is not because 211 is not equal to 0. Therefore, this subset is not a subspace of R3.
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