Mathematics Stack Exchange
Mathematics Stack Exchange

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)

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, a linear function) is a mapping $V \to W$ between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication. Reference: Wikipedia.

Linear maps can generally be represented as matrices, and simple examples include rotation and reflection linear transformations.

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Decomposition of a linear map into scale/rotation/shear matrices

When I have a linear map $$ A = \pmatrix{a & b \\ c &d} $$ It obviously has 4 degrees of freedom. I would now like to decompose it into scale, rotation and sheer components: $$ A = BCD = \pmatrix{s_1 & 0 \\ 0 & s_2}\pmatrix{\cos \alpha & -\sin…
allo
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Equivalence of surjectivity and injectivity for linear operators on finite dimensional vector spaces

I'd like to show that for a linear operator $T$ and finite-dimensional vector space $V$ such that $T:V\rightarrow V$, $T$'s injectivity is equivalent to its surjectivity. I started by trying to show $T$'s surjectivity implies its injectivity…
user2582713
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Prove $U \cap \ker T \not = 0$ where $U$ is a subspace of $\mathbf R^5 $

$\ T: \mathbf R^5 \rightarrow \mathbf R^5 , \ T^2 =0$ and let $U$ be a subspace of $\ T $ and $\dim U = 3 $ prove: $\ U \cap \ker T \not= \{0\} $ If I understand correct then $w \in \operatorname{im}T, \ v \in \mathbf R^5, \ T(v) = w,\ T(w) =…
bm1125
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Let $ T: \mathbb R^5\to \mathbb R^5$ be a linear transformation such that $T^4(x)=0$. Then find the $Rank(T)$.

Let $ T: \mathbb R^5 \to \mathbb R^5$ be a linear transformation such that $T^4(x)=0$. Then find the $Rank(T)$. Options given are a) $Rank(T)=2$ , b) $Rank(T)=3$ I dont have any idea where to begin. I posed the same question in chat in MSE, but we…
Shobhit
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Proving linearity of a function in two steps

When a function is additive, $$\color{blue}{f(x+y)=f(x)+f(y)},$$ you can extend the property to the product by an integer $$f(nx)=f(x+x+\cdots x)=f(x)+f(x)+\cdots f(x)=nf(x),$$ then to the product by a rational number $$f(\frac pqx)=\frac qqf(\frac…
user65203
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Difficulty in proving $\text{Im}f\cap\text{Ker}f\subseteqq\text{Im}(f\circ f-3f)$

I am now having difficulty in trying to prove the relation in linear mapping. $V$ is finite dimension vector space on $\mathbb{C}$, $f:V\to V$ is linear mapping. The given condition is $$\text{dim}\ \text{Im}f\cap\text{Ker}f=\text{dim}\…
Cunyi Nan
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How do I find the transformation matrix between two sets of points?

I want to turn a rectangle into a parallelogram and back, I only have the points for the two shapes. I don't care about position, just matching the shape. Is it possible to find a transformation matrix to turn a rectangle into a parallelogram given…
SpaceFace
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Proving projection map is a linear map

Let $V$ be a finite-dimensional inner product space and $W$ be a subspace of $V$. Recall that any $v \in V$ can be written as $x+y$ with $x\in W$ and $y \in W^\perp$; and that the projection $\mathrm{proj}_{W}v$ is equal to $x$ by definition. …
Twenty-six colours
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Show $T:\mathbb{R}^n\to \mathbb{R}^m$ injective, surjective then $n\leq m$,$n\geq m$ respectively

I want to show that if there is an injective linear transformation from $\mathbb{R}^n$ to $\mathbb{R}^m$ then $n \leq m$. I approached it as follows. $T$ is injective implies that the standard matrix has nullity $0$ and so rank $n$. Hence, the…
user30523
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If the base function for the transformed function $f(x)= -4(3x)^2+5$ is $f(x)=x^2$, then: is $k=3$ or $k=9$?

If the base function for the transformed function $$f(x)= -4(3x)^2+5$$ is $f(x)=x^2$, then: is $k=3$ or is $k=9$? By comparing the transformed function to: $af(k(x-d))+c$, you can pinpoint the factors which affect the function. In this case, $a=-4,…
Stephen Makanga
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Applying a transformation over a given set

While i was reading through my calculus recitations, i stumbled upon this example: suppose we have $D = \{(x,y) \in \mathbb{R^2}: 0 \leq y,0 \leq x\leq 9, 1 \leq x^2-y^2 \leq 9\}$. $U=\{(x,y) \in \mathbb{R^2} : 0
aveline de grandpre
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Linear Transformation of non-linear natural relationship in Econometrics

I'm dealing with an exercise that tells me that the natural relationship between $Y$ and two predictors $x_1$ and $x_2$ can be modelled as follows: $Y = e^{B_0+x_1B_1+x_2B_2}$/$1+e^{B_0+x_1B_1+x_2B_2}$ Then, the exercise asks me to propose a…
jamesstealth
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Need help solving this ODE with a 3x3 matrix

I have a 3x3 matrix A with the following elements: $$\begin{pmatrix} 5 & -4 & 2 \\ 4 & -5 & 4 \\ 6 & -12 & 9 \end{pmatrix}$$ The ODE is $y' = Ay$ with $y(0) = [1 \quad 0 \quad 1]^T$. I found the eigenvalue to be 3 with a multiplicity of 3. I found…
user902374
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Finding the subspace of linear transformation

Is either A or B subspace of $L(R^4,R^5)$ where $L(R^4,R^5)$ denote the set of all linear transformation form $R^4$ to $R^5$ A) $ (T\in L(R^4,R^5)|rank(T)\leq 0)$ B) the set of all $T∈L(R^5,R^4)$ such that there exist bases B and C for $R^5$ and…
PhysicsQuestion
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Dimension of the subspace of a vector space with certain conditions given.

$V$ is a finite dimensional vector space over the reals and $W$ is a subspace of $V$. I am to prove that $T(W)$ and $W$ intersect at points other than $0$ for every linear automorphism $T$ defined on $V$ if and only if $\text{dim } W > 1/2 \text{…
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