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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28 views

Linear mappings between two Sets

For $n\ge 2$ consider the following two subsets of the inner product space $\mathbb{C}^n$ with usual Euclidean inner product denoted by $<,>$ $$B=\{x\in\mathbb{C}^n:||x||<1\}$$ and, $$E=\{x\...
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What are all the linear transformations that can be done on a 2D cartesian plane?

I am interested in teaching about linear algebra on YouTube, and I have a solid grasp of the basics. However, I am looking for an exhaustive list of all the possible linear transformations of R^2 onto ...
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Help finding a linear transformation

I'm a little bit lost here; I need to find a linear transformation $T:\mathbb{R}^3\to\mathbb{R}^3$ such that $\ker(T)\cap\text{Im}(T)\neq\left\lbrace 0\right\rbrace$. I have been looking for it all ...
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Proving Linear Transformation and Kernel

Prove that L is a linear transformation and find the ker L and dim (ker L). $L: V_{4} \rightarrow P_{3}$ is defined by $L((a, b, c, d)) = ax + (a + b)x^{2} - (c + d)x^{3}$ I've gone through all my ...
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Log-log transformations exercise

I'm self-studying a book on statistics by Andrew Gelman and I'm having trouble with this exercise. I hope someone can help me here. Suppose that, for a certain population of animals, we can predict ...
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58 views

Dimension of $L(V,W)$ when $V, W$ are infinite dimensional?

For finite dimensional $V$ and $W$, we know that \begin{equation*} \dim{L(V,W)} = \dim{V}\cdot\dim{W} \end{equation*} Does this theorem hold for infinite-dimensional vector spaces $V$ and $W$ too?
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Finding vectors in orthogonal complements to create a unique sum

Take U,W to be subspaces of $\mathbb{R}^{3}$ $U = \operatorname{Lin}\left\{\left(\begin{array}{c} -1 \\ 0 \\ 1 \end{array}\right),\left(\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right)\right\}$ and $V=\...
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Why is the conjugate transpose defined like this?

If $\mathcal{A}^\dagger$ is the conjugate transpose of a linear map $\mathcal{A}$, it's defined as (at least according to my notes): $\langle u|\mathcal{A}^\dagger(v)\rangle=\langle\mathcal{A}(u)|v\...
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The linear transformation satisfies $\sigma^3=\sigma^2-2\sigma$

Suppose $\sigma$ is a linear transformtation on linear space $V$, the linear transformation satisfies $\sigma^3=\sigma^2-2\sigma$ Prove that $V=\sigma^{-1}(0)\oplus \sigma(V)$ and if $\sigma^{-1}(0)\...
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Linear Transformations from $\mathbb R$ to $\mathbb R$

Is my iff statement correct? $f:\mathbb R\to\mathbb R$ is a Linear Transformation iff there exists a unique $a\in\mathbb R$ such that for all $x\in\mathbb R$, $f(x)=xf(a)$ So if I am given any linear ...
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Given a linear transformation and basis, verify $ [T]_{\beta}^{\alpha}[v]_{\alpha} = T[(v)]_{\beta}$ . (More details in description)

Let $T : R^3 -> R^2 $ be the linear transformation defined by $T(x,y,z) = (3x +2y -4z, x-5y +3z)$, and let $\alpha$ = {(1,1,1), (1,1,0), (1,0,0)} and $\beta$ = {(1,3),(2,5)}. $Verify$ $ [T]_{\beta}...
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Change of basis to a diagonalizing one

I'm extremely confused. We generally know that for a linear transformation represented by a matrix $T$: $$P_{C}^{B}T_{B}^{B}P_{B}^{C}=T_{C}^{C}$$ And generally one can multiply a matrix $A$ for ...
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Find $\ker T^*$ (solution verification)

The problem: Let $V=\mathbb{C}^3$ with the usual inner product and $T \in L(V)$ where the matrix of the normal basis $(e_i)$ is $\begin{pmatrix} -1 &-i & 1 \\ -i &1 &i \\ 1 & i ...
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Show that $ f \circ g $ is self-adjoint iff $ f \circ g = g \circ f $ in a euclidean vector space.

This question has been asked before HERE, but I could not understand the following result from the linked post: $$ \langle F(G(v)), w \rangle = \langle G(v), F(w) \rangle = \langle v, G(F(w)) \rangle ...
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Consider the set $S = \{x : Ax = 0\}$. Show that $\dim (S) = n-r$ where $r$ is the number of Linearly Independent rows of $A$.

Consider the set $S = \{x : A_{m \times n}x = 0\}$ where $x \in \mathbb{R}^n$. Show that $\dim (S) = n-r$ where $r$ is the number of Linearly Independent rows of $A$. Can we directly use the rank-...
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What's the difference between homogeneous coordinates and projected coordinates?

I don't get how people are using homogeneous coordinates in order to construct the projection of an object. I know that homogeneous coordinates allow us to perform affine transformations in higher ...
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Rotation by an angle θ [closed]

How to find a 4x4 matrix that represents in homogeneous coordinates the rotation by an angle θ about the $p=t(1,1,1)^T+ (1,0,0)^T $ line of $R^3$ ? Thank you!
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Mapping a rectangle onto the unit square

How to find the matrix in homogeneous coordinates that maps the rectangle with vertices $(1,-2)$, $(1,2)$, $(4,2)$, $(4,-2)$ onto the unit square? I have really searched a lot but couldn't find the ...
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Hamilton-Cayley Theorem for Vector Spaces Over Any Field

In Serge Lang’s third edition of Linear Algebra on p. 243, he gives the result of the Hamilton-Cayley theorem for linear operators for vector spaces over any field $K$. My problem is that I don’t ...
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A counter for: If $f+f^{-1}$ is diagonalizable then $f$ is diagonalizable.

Let $V$ be a finite dimensional vector space and $f:V \rightarrow V$ invertible linear transformation. Prove or disprove: If $f+f^{-1}$ is diagonalizable then $f$ is diagonalizable. I know it's not ...
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Proving a vector space cannot exist

Hamilton tried to find a $3$-dimensional number system with the following properties: Every number can be written by $a + bx + cy$. This means every real number $a$ can be represented by $a + 0x + 0y$...
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Finding a flaw in a linear algebra proof

Consider $f(x, y, z) = (x^2 + 2y^2, x + z, x - z)$. Your friend wants to find the image of $f$, and their proof is the following: We know that $w$ is in the image of $f$ if and only if there is a ...
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If $T:V\longrightarrow V$ is diagonalizable, then $T$ has a characteristic polynomial of the form $f(x)=(x-c_{1})^{n_{1}}\dotsc(x-c_{k})^{n_{k}}$.

If $c_{1},\dotsc, c_{k}$ are the distinct eigenvalues of $T$ and $W_{i}$ is the eigenspace of $c_{i}$, then $T$ has characteristic polynomial of the form $f(x)=(x-c_{1})^{\dim W_{1}}\dotsc (x-c_{k})^{\...
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Are these functions linear transformations on $\mathbb R^2$? [closed]

a) $T(x,y) = (y,x)$ b) $T(x,y) = (x,\sin x)$ Trying to learn linear algebra but cant get past my difficulty with discrete mathematics
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ellipsoid and the solution of underdetermined system

Say $X$ is a fat data matrix (s by n, where each row is a sample). The relationship between $X$ and and $Y$ (s by m output matrix) is $$Y=XW$$ With pseudo-inverse, $W$ is expressed as $$X(XX^T)^{-1}Y=...
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Why this endomorphism has n distinct eigenvalues?

Let $u$ be a diagonalisable endomorphism of $\mathbb{R}^n$. Let’s suppose that the family $(\operatorname{Id}, u, u^2,..., u^{n-1})$ is linearly independent and consider $\lambda_1, \lambda_2,..., \...
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Is the rank of a linear operator over symmetric matrices preserved after rotating its domain?

I'm having a hard time trying to prove this, but somehow it seems true... The problem is: Let $\mathbb{S}^m$ be the space of all real symmetric matrices, and let $U\in \mathbb{R}^{m\times m}$ be an ...
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The matrix of an isometry has orthonormal columns

Axler's Linear Algebra Done Right proves that if $T : V \to V$ is a linear operator on a finite-dimensional inner product space over $F \in \{ \mathbb{R}, \mathbb{C} \}$, then the following are ...
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Why if $T$ is a diagonalizable linear transformation then there exists a base $B$ such that $[T]_B$ is a diagonal matrix? [closed]

In other words I don't understand why if there is a linear transformation $T$ and a base U such that $[T]_U=C^{-1}DC$ for an invertible matrix $C$ and a diagonal matrix $D$ then this means that there ...
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norm of invertible linear transformation is positive

Let $T: \mathbb{R}^n \rightarrow \mathbb{R}^n $ be invertible linear transformation, I need to show that $||T|| >0$. I think if $T$ is invertible then it has the inverse $S$ such that $TS=ST=I$ ...
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Behavior of a gas as a function of frequency by transforming the Navier-Stokes equations into frequency using the Laplace transform? [closed]

How does a gas behave as a function of frequency and not of time by transforming the Navier-Stokes equations in the frequency domain using the Laplace transform? https://en.wikipedia.org/wiki/...
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How can we state Sylvester's law of inertia without referring to a particular basis?

In elementary linear algebra, we talk about matrices, i.e. rectangular arrays of numbers. In advanced linear algebra, we prefer whenever possible to talk about abstract tensors, such as linear ...
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Linear transformation $\mathbb{P}^2 \rightarrow \mathbb{P}^2$ [closed]

Hi I dont understand how to do the following transformation Consider a transformation from $\mathbb{P}^2$ to $\mathbb{P}^2$ defined by $T(p(x)) = p(x + 3) − p(x)$ a) Proof that it is a linear ...
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24 views

Linear Transformation that maps [closed]

Let T : ℝ^2 -> ℝ^2 be a linear transformation that maps U = \begin{bmatrix}-3\\2\end{bmatrix} to \begin{bmatrix}2\\-1\end{bmatrix} and V = \begin{bmatrix}5\\-2\end{bmatrix} to \begin{bmatrix}2\\3\...
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1answer
23 views

Inequality for dimensions of eigenspaces of direct sum decomposition

I want to proof that for an endomorphism $f$ over vector space $V$ with eigenvalue $\lambda$ and $V = U \oplus V/U$ with $f(U) \subseteq U$ the following holds: $$ \dim E(f,\lambda) \leq \dim E(f_U,\...
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1answer
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Does an Orthogonal Projection map a basis of the space to a spanning set of the its subspace for a Hilbert Space? [closed]

For finite dimensional vector spaces it is quite easy to prove that a surjective linear operator $V\mapsto W$ maps a basis of $V$ to a spanning set of $W$. Is this property still true for linear ...
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Hoffman and Kunze Linear Algebra Section 6.8 Decomposition Theorem

Below you can find the related theorem and part of the proof that I am confused about. I understand that with the related definition of $E_i$'s we get k linear operators on $V$that satisfy $E_1+...+...
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Linear maps defined on pure tensors is well defined.

Suppose $V_1, \ldots, V_k, W$ are vector spaces, and suppose we define a map $f:V_1\otimes \cdots \otimes V_k \to W$ by sending pure tensor elements $v_1 \otimes \cdots \otimes v_k$ to $f(v_1 \otimes \...
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Diagonalizability of linear transformations in direct sums under quotienting

I think I have an idea of how to provide a proof for the following, but am unsure in a couple of places and would appreciate some advice: Q: Let $V$ be a finite-dimensional vector space, $U$ be a ...
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Linear Transformation - vector spaces change [closed]

Assume I have a linear transformation $T: \mathbb{R}^3 \to \mathbb{R}^3$ such that $T(x,y,z)=(y,z,x)$. Let $U=\left \{ (x,y,z) \in \mathbb{R}^3 : x+y+z=0 \right \}$ be a subspace of $\mathbb{R}^3$. I ...
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Can I generate a base of eigenvectors of a proper subspace?

I was studying linear algebra and I wondered if given a linear operator $T:V\to V$ defined in some space $V$, under which conditions one can build a base of eigenvectors for a given subspace of $V$. ...
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1answer
25 views

Finding the transformation matrix that belongs to the linear map

I am given a linear map with: $$ \begin{align} f(1,1,0) &= (3,7,1) \\ f(1,0,1) &= (3,4,2) \\ f(0,2,1) &= (-1,2,1) \end{align} $$ and I have to find a $3 \times 3$ matrix that belongs to ...
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26 views

Matrix to tensor transformation elementwise

I have a matrix $A_{ij}$ with $i,j\in[0,..,N-1]$, and I want to tranform the matrix to a tensor in the following way (Einstein notation for summation over repeated indices applies): \begin{eqnarray} ...
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Transformation matrix for projection onto hyperplane

I am given the following question here. Consider the plane $H$ in $\mathbb{R}^3$ consisting of all points satisfying the equation $x-2y+z=0$.\ a) Find an ordered basis $\mathcal{B}=\{\vec{b_1},\vec{...
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29 views

Finding the kernel of a linear transformation between polynomials

Find the kernel of the linear transformation of $T:P^4\rightarrow P^4 $ defined by $$ T(p) = p'' -p'-p$$ Previously I had to prove this was a linear transformation, and was successful, but I am having ...
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3answers
38 views

Continuity of the linear map $(x_n)_{n \in \mathbb{N}} \mapsto \sum_{k=1}^{\infty}k\cdot x_k$

Let $$E:\{(x_n)_{n \in \mathbb{N}} : \forall k \ x_k \in \mathbb{R}, \exists N\ge 0: \forall n \ge N \ x_n = 0 \} \\ L:(E,\|\cdot \|_\infty)\rightarrow\mathbb{R},\ \ \ (x_n)_{n \in \mathbb{N}} \...
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1answer
21 views

How to figure out the transformation matrix for rotation and then sheer?

I was watching this video. (Actually, I watched it 3 times because I couldn't understand it.) And right then, he showed that the trasformation matrix for rotation and then sheer is I understood how ...
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1answer
36 views

Show that $ A^t (U^⊥ )=(A^{-1} (U))^⊥ $

Let V,W be two finite dimensional vector spaces, $A:V→W$ linear and $U⊆W$ I need to show that $A^t (U^⊥ )=(A^{-1} (U))^⊥$, where $A^t:W→V$ with $(Av,w)=(v,A^t w)$ is the adjoint operator and $A^{-1}...
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2answers
34 views

Is the characteristic polynomial of a linear transformation T unique?

Say I have a linear map $T \in L(V)$. There can be multiple bases of $V$ and hence multiple matrices which represent the transformation $T$ in the form of a matrix, if we think of matrices with ...

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