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1 vote
Accepted

Biot-Savart Law Application Simplification Issues

$\newcommand{\bvec}[1]{\mathbf{#1}}$ $\newcommand{\uv}[1]{\bvec{\hat{e}}_{#1}}$ Work in the traditional Heavisideâ€“Gibbs vector calculus with cylindrical coordinates $(\rho,\phi,z)$. Let $\bvec{J}$ ...
• 8,166

Accepted

• 16k
1 vote
Accepted

Intersection of two vector parametric equations of lines

First of all, consider the definition of being perpendicular. Given your vector $\mathbf{v}$, a vector $\mathbf{u} = (a, b)^T$ is perpendicular to $\mathbf{v}$ if the inner product between the two is ...
• 264

Find subspace $T$ of $\mathbb{R}^{3}$ such that $\mathbb{R}^{3} =V \oplus T$

First determine the dimension of $V.$ The 3 vectors that generate it, are not linearly independent; if they were, then the dimension would be 3 and $T$ could only be the null space. As it is, try and ...
• 648
1 vote

Continuous basis $(e_1(t),e_2(t))$

This can be proved with elementary linear algebra. Let us write $A_t$ for $A(t)$. Let $r$ be the constant rank of all $A_t$, where $0<r<n$. (In your case $r=n-2$.) For any pair of matrices $M$ ...
• 140k
Accepted

Inverse of the identity minus a block anti-diagonal matrix.

The matrix $I-C$ is a block matrix with 2 invertible matrices on the diagonal. You can use the equation that you are linking to by substituting an $I$ of the appropriate size for $A$ and $D$, $-A$ for ...
• 648

Is every vector space an injective module?

For any vector space $M$, $$\mathrm{Ext}^1_F(M,Q)=0$$ since $M$ is free, hence projective. Thus, $Q$ is injective.
• 635

How to identify surfaces of revolution

This would be an answer for Computer Graphics/Computer Scientist folks: Check the following recent paper which provides algorithmic tools for identifying patches of surface of revolution on ...
Accepted

automorphisms on direct sum of matrices

The centre $Z$ of $M_n(R)\oplus M_m(R)$ is $\def\espan{\operatorname{span}}Z=\espan\{I_n\oplus 0, 0\oplus I_m\}$. Because $f$ is an automorphism, $f(Z)=Z$ (proof at the end). As $f(0)=0$ and $f(I)=I$, ...
• 207k