SUMMARY
The discussion focuses on calculating the value of the tensor $F(v,f)$ where $F = e^1 \otimes e_2 + e^2 \otimes (e_1 + 3e_3) \in T^1_1(V)$, with $v = e_1 + 5e_2 + 4e_3$ and $f = e^1 + e^2 + e^3$. It is confirmed that $e^1 \otimes e_2$ does not equal zero, as it corresponds to the matrix $E_{12} = \begin{bmatrix}0&1&0\\0&0&0\\0&0&0 \end{bmatrix}$, which is a non-zero matrix. The calculation of $e^i \otimes e_j(v,u^{\ast}) = u^TE_{ij}v$ yields a scalar in the underlying field, affirming the tensor's non-triviality.
PREREQUISITES
- Understanding of tensor notation and operations, specifically $T^1_1(V)$
- Familiarity with matrix representation of tensors, particularly $E_{ij}$ matrices
- Knowledge of linear algebra concepts, including vector spaces and dual spaces
- Proficiency in manipulating linear combinations of basis vectors
NEXT STEPS
- Study the properties of tensor products in linear algebra
- Learn about the application of matrices in representing tensors
- Explore the implications of dual spaces in tensor calculations
- Investigate the significance of non-zero tensors in vector space theory
USEFUL FOR
Mathematicians, physicists, and students studying linear algebra and tensor analysis, particularly those interested in tensor products and their applications in various fields.