Part C4: Tensor product
... to another exact sequence of the same form: M ⊗ A → M ⊗ B → M ⊗ C → 0. This statement appears stronger than the original statement since the hypothesis is weaker. But I explained that the first statement implies this second version. Suppose that we know that M ⊗ − sends short exact sequences to righ ...
... to another exact sequence of the same form: M ⊗ A → M ⊗ B → M ⊗ C → 0. This statement appears stronger than the original statement since the hypothesis is weaker. But I explained that the first statement implies this second version. Suppose that we know that M ⊗ − sends short exact sequences to righ ...
Minimal spanning and maximal independent sets, Basis
... Let S be a set of real n-vectors. In particular, S can be a linear space, or its subspace. Yet, S can also be just a finite set of vectors, S = {x1 , ..., xm }, where xi = (xi1 , ..., xin ) and i = 1, ..., m. For example S is the set of columns of an (m × n) real matrix A. Given k vectors x1 , ..., ...
... Let S be a set of real n-vectors. In particular, S can be a linear space, or its subspace. Yet, S can also be just a finite set of vectors, S = {x1 , ..., xm }, where xi = (xi1 , ..., xin ) and i = 1, ..., m. For example S is the set of columns of an (m × n) real matrix A. Given k vectors x1 , ..., ...
SageManifolds: A free tool for differential geometry and
... f.express = C : F, Ĉ : F̂ , . . . with f ( p ) = F ( x1 , . . . , xn ) = F̂ ( x̂1 , . . . , x̂n ) = . . . | {z } |{z} | {z } point coord. of p coord. of p in chart C in chart Ĉ ...
... f.express = C : F, Ĉ : F̂ , . . . with f ( p ) = F ( x1 , . . . , xn ) = F̂ ( x̂1 , . . . , x̂n ) = . . . | {z } |{z} | {z } point coord. of p coord. of p in chart C in chart Ĉ ...
1 Eigenvalues and Eigenvectors
... 10. Later in Chapter 5, we will find out that it is useful to find a set of linearly independent eigenvectors for a given matrix. The following theorem provides one way of doing so. See page 307 for a proof of this theorem. 11. Theorem 2: If v1 , . . . , vr are eigenvectors that correspond to distin ...
... 10. Later in Chapter 5, we will find out that it is useful to find a set of linearly independent eigenvectors for a given matrix. The following theorem provides one way of doing so. See page 307 for a proof of this theorem. 11. Theorem 2: If v1 , . . . , vr are eigenvectors that correspond to distin ...
Matrix manipulations
... We say a matrix A ∈ Rn×m is data sparse if we can represent it with far fewer than nm parameters. For example, • Sparse matrices are data sparse – we only need to explicitly know the positions and values of the nonzero elements. • A rank one matrix is data sparse: if we write it as an outer product ...
... We say a matrix A ∈ Rn×m is data sparse if we can represent it with far fewer than nm parameters. For example, • Sparse matrices are data sparse – we only need to explicitly know the positions and values of the nonzero elements. • A rank one matrix is data sparse: if we write it as an outer product ...
Momentum and Impulse
... Steve kicks a ball of mass 0.8kg along the ground at a velocity of 5ms-1 towards Monica. She kicks it back towards him, but lofts it so that it leaves her foot with a speed of 8ms-1 and with an elevation of 40° to the horizontal. Find the magnitude and direction of the impulse of Monica’s kick. ...
... Steve kicks a ball of mass 0.8kg along the ground at a velocity of 5ms-1 towards Monica. She kicks it back towards him, but lofts it so that it leaves her foot with a speed of 8ms-1 and with an elevation of 40° to the horizontal. Find the magnitude and direction of the impulse of Monica’s kick. ...
