# Tagged Questions

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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### Operatornorm of $(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$

Determine the operatornorm of the mapping $I:(\mathbb{R}^d, \|\cdot\|_1) \to (\mathbb{R}^d, \|\cdot\|_{\infty})$! Unfortunately I haven't many ideas for this task. I know that the definition of the ...
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### Construct piecewise formula for position s, given accel position, velocity

I'm writing an algorithm to control the position of a motorised system, and I'm trying to construct a formula which I can then translate into C. I'd like some examples of piecewise formulas which ...
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### Baker-Campbell-Hausdorff/Zassenhaus formula to first order in one matrix

Is there a closed-form expression for the term of $e^{t(c \hat{X} + d \hat{Y})}$ that is first-order in $d$, where $t$, $c$, and $d$ are scalars and $\hat{X}$ and $\hat{Y}$ are finite-dimensional ...
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### Implementing gradient descent based on formula

The gradient descent algorithm is given as : repeat { $$\displaystyle \theta_j := \theta_j - \frac{1}{m} \alpha \sum_{i=1}^m (h_\theta(x^{(i)}) - y^{(i)}) x^{(i)}_j$$ } Given these values : <...
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### Whether a given algebra is the algebra of endomorphisms for a vector space.

Let $\mathbb{F}$ be a field and let $A$ be an associative unital $\mathbb{F}$-algebra. Is there a criterion to let me know if $A$ is isomorphic to the algebra $\mbox{End}(\mathbf{V})$ of endomorphisms ...
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### How do I compute the area of this parallelogram

Given vectors $a,b$ and the ribs of parallelogram are $2a +3b = A$, $a-2b = B$. Also given $a \times b = (-1,2,2)$. Compute the surface of the parallelogram. I'm not sure where I saw but I think it ...
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### Showing properties of a function and its inverse image

I tried proving the following question but did not get too far. Let $\ f:A \to B$ be a function and $\ f^{-1}(Y)$ be the inverse image of $\ Y\subseteq B$ on $\ f$. Consider the following ...
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### commutativity of log(I + A) and log( A−1) (matrix function)

I'm self-(re)learning linear algebra since the beginning of the summer, and i have a problem with the following exercice entitled additive logarithmic. If i'm right, we need to prove the ...
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### Is this an Example of a Dual Space? [on hold]

Is the set of possible bases that I describe $∀(e_1,e_2,e_3)$justSlash$∀(e_1,e_2,F(e_1, e_2))$ F defined V, \times. v=e_1 \times e_2*for any linear vector space of dimension 3* and their linear ...
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### Examples for proof of geometric vs. algebraic multiplicity

Here you see a supposedly easy proof of a well-known theorem in linear algebra: Although I know I should understand this, I don't :-( Obviously there are too many indices and stuff, so I don't see ...
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### Determine point of interesction of plane with axis given points of plane

Q: The points $(2,-1,-2)$, $(1,3,12)$ and $(4,2,3)$ lie on a unique plane. Where does the plane cross the z-axis. I understand that the point of intersection would occur at $(0,0,z)$ and I have to ...
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### Don't know how to enter this into webwork [on hold]

I know the vectors are (-3-i ; 2) and (-3+i ; 2) however no matter which way I enter it into the program, it regards my answer as incorrect. How am I to enter the answer?
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### What is the nullity of an onto transformation?

For a $5 \times 13$ matrix, with $T(x) = Ax$, what is the nullity of $A$ if $T$ is onto? I can't figure out what it would be...
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### Smallest possible value of the norm?

The vectors $\vec{u_1} = \begin{bmatrix} 1 \\ 1 \\ 1\\ 1 \end{bmatrix}$ and $\vec{u_2} = \begin{bmatrix} 1 \\ -1 \\ 1\\ -1 \end{bmatrix}$ are orthonormal in $\mathbb{R}^4$....
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### Gradient and Hessian of function on matrix domain

Let $A \in R^{k \times p}$. Define $f(X) : R^{p \times k} \rightarrow R$ to be $f(X) = \log \det(XA + I_{p})$, where $I_{p}$ is a $p \times p$ identity matrix. I want to know what is the gradient and ...
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### Completing a semi-vector space to a vector space

If we have a semi-vector space $U$ (as defined here), what do we have to additionally demand from $U$ such that we can complete it to a vector space $\tilde{U}$ via $U\xrightarrow{\iota}\tilde{U}$ and ...
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