Linear algebra is concerned with vector spaces of all dimensions and linear transformations between them, including systems of linear equations, bases, dimensions, subspaces, matrices, determinants, traces, eigenvalues and eigenvectors, diagonalization, Jordan forms, etc. For questions specifically ...

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Intuition behind functional dependence

What is the intuition behind functional independence ? (This is defined in the following way: Let $k\leq n$. The $C^1$ functions $F_1,\ldots,F_k:\mathbb{R}^n\rightarrow \mathbb{R}$ are functionally ...
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Lang's Linear Algebra: what's next?

I've completed the study of Lang's Linear Algebra ($3^\text{rd}$ edition). To put it simply, I have enjoyed the subject and I would like to know "what's next". In other words, I would like to know ...
5
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2answers
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(still open) For what $X$ do we have $\|XA\|_{1} \leq 1$ for a given $\|A\|_{1} \leq 1$.

All matrices are real. By $\| \cdot \|_1$ I mean a matrix norm induced by the vector norm $L_1$, i.e. the max of the column sums of absolute values. The matrix $A$ is given and we have $\|A\|_{1} ...
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1answer
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I cannot make the mental leap from a vector to a function!

In my linear algebra book, it says that a vector is linearly independent if $\vec V = c1*\vec T_1 + c2*\vec T_2$ And then it goes on to say that $y(t) = c1 * e^{-at} + c2*e^{-bt}$ is linearly ...
5
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0answers
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Is there a generalization of the Lagrange polynomial to 3D?

What is a way to construct a smooth polynomial surface ($\mathbb{R}^2 \rightarrow \mathbb{R}$) with Lagrange-polynomial properties in every partial derivative? I want to try this for image ...