SUMMARY
The discussion centers on determining whether specific sets of vectors are bases for the complex vector space C^3. The first set, {(i,0,-1),(1,1,1),(0,-i,i)}, is confirmed as a basis because it contains three linearly independent vectors, as evidenced by having three leading entries in the corresponding matrix. In contrast, the second set, {(i,1,0),(0,0,1)}, is not a basis for C^3 since it contains only two vectors, which is insufficient to span the three-dimensional space. The conclusion is that a basis for C^3 must consist of exactly three linearly independent vectors.
PREREQUISITES
- Understanding of vector spaces and their dimensions
- Knowledge of linear independence and leading entries in matrices
- Familiarity with complex numbers and their representation as vectors
- Ability to perform matrix operations and row reduction
NEXT STEPS
- Study the concept of linear independence in vector spaces
- Learn about the properties of bases in vector spaces
- Explore matrix row reduction techniques for determining independence
- Investigate the implications of dimensionality in complex vector spaces
USEFUL FOR
Students and educators in linear algebra, mathematicians focusing on vector spaces, and anyone interested in the properties of complex numbers in higher dimensions.