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Show that \(\operatorname{trace}(A B C)=\operatorname{trace}(C A B)=\operatorname{trace}(B C A)\).
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Show that \(\operatorname{trace}(A B C)=\operatorname{trace}(C A B)=\operatorname{trace}(B C A)\).Let \(A, B\), and \(C\) be any \(n \times n\) matrices. a) Show that \(\operatorname{trace}(A B)=\operatorname{trace}(B A)\). b) Show that \(\operator ...
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asked in Mathematics 19 minutes ago
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Find the dimension of \(A\) considered as a real vector space.
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Find the dimension of \(A\) considered as a real vector space.Consider the two linear transformations on the vector space \(V=\mathbf{R}^{n}\) : \(R=\) right shift: \(\left(x_{1}, \ldots, x_{n}\right) \rightarrow ...
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asked in Mathematics 37 minutes ago
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Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a real matrix, not necessarily square.
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Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a real matrix, not necessarily square.Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a real matrix, not necessarily square. a) If two columns of \(A\) are the same, show that \(A ...
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asked in Mathematics 9 hours ago
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Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map. Show that the following are equivalent.
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Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map. Show that the following are equivalent.Let \(A: \mathbb{R}^{n} \rightarrow \mathbb{R}^{k}\) be a linear map. Show that the following are equivalent. a) For every \(y \in \mathbb{R}^{k}\) th ...
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asked in Mathematics 12 hours ago
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Let \(C(\mathbb{R})\) be the linear space of all continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\).
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Let \(C(\mathbb{R})\) be the linear space of all continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\).Let \(C(\mathbb{R})\) be the linear space of all continuous functions from \(\mathbb{R}\) to \(\mathbb{R}\). a) Let \(S_{c}\) be the set of different ...
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asked in Mathematics 13 hours ago
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The five-digit number 1234A is divisible by 9. What is the remainder when it is divided by 7 ?
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Determine \(\frac{\mathrm{d} y}{\mathrm{~d} x}\) if \[ y=\left(x^{2}+\frac{1}{x^{2}}\right)^{2} \]
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Given \[ y=\sqrt{x^{7}}-\frac{5}{x^{3}} \] Which of the following is equivalent to \(y\) ?
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Given \[ y=\sqrt{x^{7}}-\frac{5}{x^{3}} \] Which of the following is equivalent to \(y\) ?Given \ y=\sqrt{x^{7}}-\frac{5}{x^{3}} \ Which of the following is equivalent to \(y\) ? \begin{align} \begin{array}{|l|l|} \hline \text { A } &amp ...
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asked in Mathematics Jan 15
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Differentiate \(g(t)=4(t+1)^{2}(t-3)\) with respect to \(t\)
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Differentiate the function \(m=\frac{3}{2} n^{4}-1\)
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Differentiate the function \(y=3 x^{5}\)
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\begin{equation} \text { Consider the system } A X=B, \text { where } X=\left[\begin{array}{l} x \\ y \\ z \end{array}\right] \text {. } \end{equation}

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The function $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$ has
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The function $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$ has The function $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$ has   A. no extremum, B. exactly one maximum, C. exactly one minimum, D. exactly ...
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asked in Mathematics Jan 13
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The function $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$ is
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The function $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$ isThe function $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$ is   A. strictly increasing on $(-\infty, \infty)$,   B. strictly increasing on $(-\infty ...
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asked in Mathematics Jan 13
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The function $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$ has ...
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The function $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$ has ...The function $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$ has   A. no point of inflection and is concave up on $(-\infty, \infty)$,   B. no point o ...
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asked in Mathematics Jan 13
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Prove Rolle's Theorem under the assumption that for the function $f$ on $[a, b]$ you have $f(x)>f(a)$ for some $x$ in the domain of $f$.
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Sketch the graph of $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$
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Sketch the graph of $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$Sketch the graph of $f(x)=\dfrac{x^{3}-6 x}{x^{2}-4}$ ...
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asked in Mathematics Jan 13
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Evaluate the integral: $\int \dfrac{3 x+2}{\sqrt{4-3 x^{2}}} d x$
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Evaluate the integral: $\int \dfrac{3 x+2}{\sqrt{4-3 x^{2}}} d x$Evaluate the integral:   $\int \dfrac{3 x+2}{\sqrt{4-3 x^{2}}} d x$ ...
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asked in Mathematics Jan 13
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Evaluate the integral: $\int_{0}^{\pi} 2 x \cos 2 x d x$
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Evaluate $\int \dfrac{\ln z}{(1+z)^{2}} d z$
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Evaluate the following integral: $\int \frac{t}{\sqrt{t^{4}-1}} d t$.
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Evaluate the following integral: $\int \frac{t}{\sqrt{t^{4}-1}} d t$. Evaluate the following integral: $\int \dfrac{t}{\sqrt{t^{4}-1}} d t$. Hint: You may use the fact that $\int \sec \theta d \theta=\ln |\sec ...
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asked in Mathematics Jan 13
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Evaluate the integral $\int_{1}^{2} \dfrac{2 s-2}{\sqrt{-s^{2}+2 s+3}} d s$
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Evaluate $\int \dfrac{1}{\left(1+x^{2}\right) \arctan x} d x$
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