# Questions tagged [linear-transformations]

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping V → W between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. (Def: http://en.m.wikipedia.org/wiki/Linear_map)

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### Convert trapezoidal velocity profile into one with s-curve

I am an electronic engineer. I have created a system that contains a stepper motor. There is a speed table that specifies the speed values that are to be used to accelerate and deaccelerate the motor. ...
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### Preserving Leading Eigenvalue while Shrinking Matrix

Suppose I have a graph adjacency matrix $G$ of size $n \times n$. By Perron-Frobenius, I am guaranteed to have a unique largest real eigenvalue $\lambda_{G,1}$ with corresponding eigenvector $u_{G,1}$...
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### Is it possible that linear transform changes interval data to ratio?

I'm going through a stats intro and got puzzled by the concept of interval and ratio data. Celsius temps is an example of interval data that can not be multiplied and divided, because there is no true ...
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### If $P_1$ and $P_2$ are orthogonal projectors, then $\mathrm{tr}(P_1 P_2) \leq \mathrm{rk}(P_1 P_2).$

Prove or provide a counterexample: If $P_1$ and $P_2$ are orthogonal projectors, then $\mathrm{tr}(P_1 P_2) \leq \mathrm{rk}(P_1 P_2),$ where $\mathrm{tr}$ denoted the trace and $\mathrm{rk}$ the ...
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Can someone provide Hint to Prove this. Let $S: R^2 \rightarrow R^2$ be linear, such that $S^2 = S$. Prove that $S= 0$, $S=id_{R^2}$ or there exists B a basis of $R^2$ such that $[S]^B_B=\begin{... • 1 2 votes 1 answer 73 views ### True or False: There is a$6\times 6$matrix$A$with$\text{Rank}(A)=4$and$A^3 =0$I understand how to do it if the question changed$A^3$to$A^2$, because then you can just use the rank–nullity theorem.$\text{Rank}+\text{Nullity}=6$,$\text{Rank}=4$so$\text{Nullity}=2$so of ... • 21 0 votes 1 answer 42 views ### A isomorphism between$\mathcal{L}(U,V)$and$\mathcal{L}(U) \times \mathcal{L}(V)$. Let$U$,$V$be two Banach spaces and define the spaces $$\mathcal{L}(U\times V) = \{T : U\times V \rightarrow U\times V : T \text{ is linear and bounded}\},$$$$\mathcal{L}(U) = \{T : U \... 2 votes 1 answer 72 views ### Prove whether the following transformation is linear or not. I'm not sure if I'm going wrong with the proof, but I'm getting my answer as, the transformation is linear, whereas according to our teacher, it is not supposed to be linear. Q.$T(v_1, v_2) = (v_1, ...
My question is as follows. Let $T : V \to V$ be a linear map of a finite dimensional vector space V. If $ker T = Im T$, then is $dim V$ even and $T^{2} = 0$? I believe this to be false since I have ...