### Questions (52)

 8 How do we know that the number $1$ is not equal to the number $-1$? 7 Spectral theorem/ exchanging limit of series and operator 6 Is there a universal property of tangent space just like in the case of the tensor product? 5 Let $A_1,A_2\in L(V,V)$, $A_i(V)=V$ and $\dim\ker A_i=n_i\in\mathbf{N}$. Is $\dim\ker (A_2A_1)=n_1+n_2$ even if $\dim V=\infty$? 4 Tensor product vector bundle - bijection from vector valued functions to sections defined by local sections

### Reputation (952)

 +10 Trace of operator defined on direct sum of two vector spaces +10 Definition of the twisted covariant derivative +2 Basis for tensor products +10 Got $e^{-x}=\cos(ix)$ from Euler's formula. Where is my mistake?

### Answers (26)

 6 How to tell if an image of a triangle has half its area shaded? 4 Is there a universal property of tangent space just like in the case of the tensor product? 2 Got $e^{-x}=\cos(ix)$ from Euler's formula. Where is my mistake? 1 Why $\{\mathbf{0}\}$ has dimension zero? 1 According to what rigorous definition of “polynomial” is $\sum_{k=0}^n a_k D^k$ a polynomial, where $D$ is the derivative operator?

### Tags (86)

 6 geometry × 3 2 complex-numbers × 5 4 manifolds × 9 2 algebra-precalculus 4 tangent-spaces × 5 2 exponential-function 4 universal-property × 2 2 trigonometry 2 linear-algebra × 15 1 reference-request × 4

### Bookmarks (11)

 8 Non-degenerate bilinear forms and invertible matrices 8 Tensor fields and vector bundles 6 Why are $C^\infty_p\neq C^\infty_q$ when $p\neq q$? 4 Calculating the divergence of a vector field using non-cubic volumes 4 Is the tangent bundle the DISJOINT union of tangent spaces?

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