# 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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### How to find a matrix representing the linear transformation $xp(x)\mapsto (x-1) p(x)$?

Does there exist a matrix representation for the linear transformation $T(x P(x)) = (x-1)P(x)$, where $P(x)$ is the second degree polynomial? Here, $xP(x)$ are all third degree polynomials that ...
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### Linear Algebra--searching a name for certain transformations

I am currently taking a Linear Algebra class in Spanish and having difficulty coming across the correct translation for what we are studying. I am looking at a question that asks for the rotation of ...
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### If $T: X\to Y$ is a bijective closed linear operator, show that $T^{-1}$ is bounded.

Let $X$ and $Y$ be normed spaces and $X$ compact. If $T: X\to Y$ is a bijective closed linear operator, show that $T^{-1}$ is bounded. I don't know where to start here. Any help would be appreciated....
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### Kernel and Range of a linear transformation

So the question is let T:M2x2 -> R be defined by T(A) = tr(A). Find bases for the kernel and range of the linear transformation T. Could someone explain how to solve this as I don't quite understand ...
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### What exactly is Standard Coordinates?

What exactly is a standard coordinates? Sorry it seems like a very stupid question, but my professor didn't really explain it and just started to use it for solving other problems related to ...
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### $r(S+T)\le r(S) + r(T)$?

If $V$ is a finite-dimensional vector space, and $S$ and $T$ are linear transformations from $V$ to $V$, how can you show that $\text{im}(S+T)$ is a subset of $\text{im}(S) + \text{im}(T)$ and also ...
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### If a composition of linear transformation is invertible, then are each linear transformations invertible?

If $S$ and $T$ are linear transformations from set $V$ to $V$, which is a finite-dimensional vector space, and if the composition $ST$ is invertible, how can we show that $T$ is one-to-one, therefore, ...
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### Determine matrix of linear transformation

Let $T:R^2\rightarrow R^2$ by $$T \left( \begin{bmatrix} x_{1} \\ x_{2}\end{bmatrix} \right) = \begin{bmatrix} x_{2} \\ x_{1}\end{bmatrix}$$ Let A be the matrix of T. What is A. I'm having trouble ...
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### Computing Hermitian Conjugate for an Operator on a Function

The operator $\hat D$ is defined by $(\hat D f)(x) = \sqrt 2 f(2x)$. Show that $\hat D$ is a linear transformation, compute its hermitian conjugate and show it is unitary. Determine all eigenfunctions ...
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### linear transformation between Hilbert space

By definition, $|T|=\sup|(Tf,g)|, |f|\le1,|g|\le1$ $$||T||\ge(Tf,f)$$ But I can not find an example such that $||T||>(Tf,f)$ for any $|f|<1$. Any suggestion? Thanks in advance~
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### Validity of this geometry proof

In triangle ABC, D is the midpoint of AB and E is the point of trisection of BC nearer to C. Given that $\angle$ADC = $\angle$ BAE find $\angle$BAC. Source Q5 Lets call the intersection ...
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### Proof of Linear transformation

Let $T$ be a linear transformation from $\mathbb{R}^n$ to itself. For a given vector $v$ of $\mathbb{R}^n$, if $T(v)\neq 0$ but $T^2(v) = T(T(v)) = 0$, then prove that $v$ and $T(v)$ are linearly ...
I have a homework problem from Hoffman and Kunze's Linear Algebra. Let F be a subfield of $\mathbb{C}$ and let $T$, $D$ be the transformations of $F[x]$ defined by \begin{align} T\left(\sum_{i=0}^n{... 1answer 862 views ### Linear transformation with clockwise rotation on z axis Let T be a linear Transformation from \mathbb{R}^3 to itself such that T is 60^{\circ} clockwise rotation with fixed z-axis (i.e, rotate the space according to the z-axis) where \mathbb{R}... 1answer 92 views ### Proving linear transformation and that T is the differential operator I have two problems that are pretty short. I understand the concepts behind, however I am not sure if my proofs are insufficient: Determine whether T: M_{nn}\to\mathbb{R} defined by T(A)=a_{11}a_{... 1answer 378 views ### Kernel, Range, and Matrix Representation of a Linear Transformation Let L be defined on P_3 (the vector space of polynomials of degree less than 3) by L(p) = q where q(x) = 4p(x) − 3xp'(x) + x^2 p''(x). (a) Find the range of L in the form Span(. . .). (b) Find ... 1answer 289 views ### Finding a specific camera transformation matrix I have the following situation: - two targets with known coordinates with respect to the "world". They are on a fixed xy plane on a height 0 in the z-direction. - Both targets have an angle associated ... 3answers 1k views ### If I-AB is invertible, then is I-BA invertible? [duplicate] If A, B are square matrices and I-AB is invertible how do I prove that I-BA is invertible? This is exercise 8 of section 6.2 in Linear Algebra by Hoffman and Kunze. My thoughts. If A and ... 1answer 96 views ### Reducing a matrix using similarity transformations I'm trying to reduce a matrix to an Upper Hessenberg form with similarity transformations. I figured that the Householder Method would be the way to solve this problem, but I'm having problems with ... 1answer 22 views ### Co-ordinate vector of the linear transformation of x T is the linear transformation of V (n-dimensional) to W (m-dimensional) and {b_1,...b_n} is the basis B for V. Given any x in V, the coordinate vector [x]_B is in R^n and the ... 1answer 123 views ### When restriction of a diagonalizable linear operator to an eigenspace is also diagonalizable ? Let T, S be diagonalizable linear operators on \mathbb R^n such that TS = ST. Let E be an eigenspace of T. Is it true that the restriction of S to E is diagonalizable ? 1answer 35 views ### T:\mathbb C^n \to \mathbb C^n is linear and \ker(T-aI)=\ker(T-aI)^n , \forall a\in \mathbb C , then T diagonalizable ? If T:\mathbb C^n \to \mathbb C^n is a linear transform such that \ker(T-aI)=\ker(T-aI)^n , \forall a\in \mathbb C , then is T diagonalizable ? 1answer 52 views ### How to calculate the Matrix of a given Linear Transformation? Let V = F^3 and W = F^4 and we define the following functions: p\in {\cal L}(V,F) given by p((x,y,z)) = 3x + 4y + 2z q\in {\cal L}(W,F) given by q((w,x,y,z)) = 2w + 5x + 7y + 11z; T\in ... 2answers 45 views ### Linear transformation and vectors on a certain type of 3 dimensional vector space over the rational number field Let V be a 3 dimensional vector space over \mathbb Q and T be a linear transform on V such that for some \vec x , \vec y , \vec z \in V with \vec x \ne \vec 0 , T(\vec x)=\vec y , T(\... 1answer 41 views ### Linear Transformation: When T(v)=v I have a homework to do in which with a pre-defined transform I have to find a vector v that after the transformation equals itself: T(v)=v. The transformation happens from \mathbb{R}^3 to \... 2answers 617 views ### What do we mean by Derivative of linear function is a constant function. I've the text below given in my notes: Derivative of linear function: Let R:X\to Y be a linear function .Then R':X\to L(X,Y) is a constant function with the constant value R\in L(X,Y) i.e. ... 0answers 70 views ### If \overline f=f-f'(a) then how is \overline {f'(a)}=0? Below is the definition of a function being differentiable at a point, given in my notes: A function f:A \rightarrow Y is said to be differentiable at a \in A if there is a linear map T \in ... 1answer 279 views ### Let a_1, …,a_n , b_1,…b_n be 2n distinct elements of a field , then is the matrix \Big(\dfrac1{a_i-b_j}\Big)_{ij} non-singular? Let a_1, ...,a_n , b_1,...b_n be 2n distinct elements of a field and defineh_{ij}:=\dfrac1{a_i-b_j} , \forall i,j=1,2 ,\dots,n. $$Is the n \times n matrix H:=(h_{ij}) non-singular ? 1answer 673 views ### Matrix of Shift Transform on arbitrary basis This is problem 4.4.8 of Algebra by Artin. Let V be a vector space with basis (v_1,...,v_n) over a field F, and let a_1,...,a_{n-1} be elements of F. Define a linear operator on V by the ... 2answers 2k views ### Find a matrix transformation mapping \{(1,1,1),(0,1,0),(1,0,2)\} to \{(1,1,1),(0,1,0),(1,0,1)\} Find a matrix transformation mapping \{(1,1,1),(0,1,0),(1,0,2)\} to \{(1,1,1),(0,1,0),(1,0,1)\}. Is the answer$$ \begin{bmatrix}1& 0& -1\\0& 1& 1\\0& 0& 1\end{bmatrix}? ...
I have a doubt in the following question. Suppose $T(\vec v) = \vec v$, except that $T(0, v_2) = (0, 0)$. Show that this transformation satisfies $T(cv) = cT(v)$ but not $T(v + w) = T(v) + T(w)$. I ...