Question

For each of the following sets, indicate whether it is a vector space. If so, point out a basis of it; otherwise, point out which vector-space property is violated.

1.The set V of vectors [2x, x2] with x R2. Addition and scalar multiplication are defined in the same way as on vectors.

2.The set V of vectors [x, y, z] R3 satisfying x + y + z = 3 and x − y + 2z = 6. Addition and scalar multiplication are defined in the same way as on vectors.

3. The set V of symmetric 2 × 2 matrices. Addition and scalar multiplication are defined in the same way as on matrices.
4. The set V of 2 × 2 matrices [aIJ ] with a11 + a22 = 0. Addition and scalar multiplication are defined in the same way as on matrices.

Answer 1

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