1. ### I Trying to get a better understanding of the quotient V/U in linear algebra

Hi! I want to check if i have understood concepts regarding the quotient U/V correctly or not. I have read definitions that ##V/U = \{v + U : v ∈ V\}## . U is a subspace of V. But v + U is also defined as the set ##\{v + u : u ∈ U\}##. So V/U is a set of sets is this the correct understanding...
2. ### Question about linear transformations

Summary:: linear transformations Hello everyone, firstly sorry about my English, I'm from Brazil. Secondly I want to ask you some help in solving a question about linear transformations. Here is the question: Consider the linear transformation described by the matrix \mathsf{A} \in \Re...
3. ### I Linear transformations

Let A={ex,sin(x),excos(x),sin(x),cos(x)} and let V be the subspace of C(R) equal to span(A). Define T:V→V,f↦df/dx. How do I prove that T is a linear transformation? (I can do this with numbers but the trig is throwing me).
4. ### I Find matrix of linear transformation and show it's diagonalizable

The strategy here would probably be to find the matrix of ##F##. How would one go about doing that? Since ##V## is finite dimensional, it must have a basis...
5. ### I Proof of ##F## is an orthogonal projection if and only if symmetric

The given definition of a linear transformation ##F## being symmetric on an inner product space ##V## is ##\langle F(\textbf{u}), \textbf{v} \rangle = \langle \textbf{u}, F(\textbf{v}) \rangle## where ##\textbf{u},\textbf{v}\in V##. In the attached image, second equation, how is the...
6. ### Projections and direct sum

Homework Statement Let ##V = \mathbb{R}^4##. Consider the following subspaces: ##V_1 = \{(x,y,z,t)\ : x = y = z\}, V_2=[(2,1,1,1)], V_3 =[(2,2,1,1)]## And let ##V = M_n(\mathbb{k})##. Consider the following subspaces: ##V_1 = \{(a_{ij}) \in V : a_{ij} = 0,\forall i < j\}## ##V_2 =...
7. ### Eigenvectors and orthogonal basis

Homework Statement I have a linear transformation ##\mathbb{R}^3 \rightarrow \mathbb{R}^3##. The part that asks for a basis of eigenvectors I've already solved it. The possible eigenvectors are ##(1,-3,0), (1,0,3), (\frac{1}{2}, \frac{1}{2},1) ##. Now the exercise wants me to show that there is...
8. ### Linear transformation representation with a matrix

Homework Statement For the linear transformation T: R2-->R2 defined by T(x1, X2) = (x1 + x2, 2x1 - x2), use the matrix A to find T(v), where v = (2, 1). B = {(1, 2), (-1, 1)} and B' = {(1, 0), (0, 1)}. Homework Equations T(v) is given, (x1+x2, 2x1-x2) The Attempt at a Solution Okay, I see...
9. ### Matrix of linear transformation

Homework Statement Let A:\mathbb R_2[x]\rightarrow \mathbb R_2[x] is a linear transformation defined as (A(p))(x)=p'(x+1) where \mathbb R_2[x] is the space of polynomials of the second order. Find all a,b,c\in\mathbb R such that the matrix \begin{bmatrix} a & 1 & 0 \\ b & 0 & 1 \\ c & 0...
10. ### I What [T]_gamma signifies?

let's consider we have a linear transformation T: R^2->R^3 and alpha={ordered basis of R^2} and beta{ordered basis of R^3} and gama={v1,v2}, v1=(1,-1),v2=(2,-5). now I need to find [T]_gama(associated matrix)? When i read about it, i understood it as, first we have to find transformation of each...
11. ### I Linear Transformation notation

I'm confused about the notation T:R^n \implies R^m specifically about m. From my understanding if n=2 then (x1, x2). Are we transforming n=2 to another value m for example (x1, x2, x3)?
12. ### Linear algebra: Find the matrix of linear transformation

Homework Statement Check if L(p)(x)=(1+4x)p(x)+(x-x^2)p'(x)-(x^2+x^3)p''(x) is a linear transformation on \mathbb{R_2}[x]. If L(p)(x) is a linear transformation, find it's matrix in standard basis and check if L(p)(x) is invertible. If L(p)(x) is invertible, find the function rule of it's...
13. ### Linear Transformation and isomorphisms

Homework Statement Suppose a linear transformation T: [P][/2]→[R][/3] is defined by T(1+x)= (1,3,1), T(1-x)= (-1,1,1) and T(1-[x][/2])=(-1,2,0) a) use the given values of T and linearity properties to find T(1), T(x) and T([x][/2]) b) Find the matrix representation of T (relative to standard...
14. ### Linear Algebra - Transformation / operator

Homework Statement Let T:V→V be a linear operator on a vector space V over C: (a) Give an example of an operator T:C^2→C^2 such that R(T)∩N(T)={0} but T is not a projection (b) Find a formula for a linear operator T:C^3→C^3 over C such that T is a projection with R(T)=span{(1,1,1)} and...
15. ### Derivatives and Linear transformations

Let G be a non-empty open connected set in Rn, f be a differentiable function from G into R, and A be a linear transformation from Rn to R. If f '(a)=A for all a in G, find f and prove your answer. I thought of f as being the same as the linear transformation, i.e. f(x)=A(x). Is this true?
16. ### Linear algebra problem related to vector subspace

Homework Statement X ={(x1,x2,x2 −x1,3x2):x1,x2 ∈R} f(x1,x2,x2 −x1,3x2)=(x1,x1,0,3x1) 1. Find a basis for X. 2. Find dim X. 3. Find ker f and im f 4. Find bases for ker f and im f 5. Is f a bijection? Why? 6. Find a diagonal matrix for f. Homework Equations The Attempt at a Solution 1. Put...
17. ### Vector subspace and linear transformation

X ={(x1,x2,x2 −x1,3x2):x1,x2 ∈R} f(x1,x2,x2 −x1,3x2)=(x1,x1,0,3x1) 1. Find a basis for X. 2. Find dim X. 3. Find ker f and im f 4. Find bases for ker f and im f 5. Is f a bijection? Why? 6. Find a diagonal matrix for f. My attempt: 1. (1, 1, 0, 3) and (1, 2, 1, 6) 2. Dim X = 2 3. Ker f = 0, im...
18. ### Finding properties of a linear transformation

Homework Statement Find the domain, target space, image, rank and nullity of the linear transformation T(A)=Av, where v= (1, 2) and A is any 2×2matrix. Homework Equations The Attempt at a Solution I believe I know the domain (R2x2 vector space) and target space (R2), but I am not sure how to...
19. ### Linear Transformations and Image of a Matrix

Homework Statement Consider a 2x2 matrix A with A2=A. If vector w is in the image of A, what is the relationship between w and Aw? Homework Equations Linear transformation T(x)=Ax Image of a matrix is the span of its column vectors The Attempt at a Solution I know that vector w is one of the...
20. ### Need help finding the preimage

Consider the linear transformation T: R3 --> R3 /w T(v1,v2,v3)=(0, v1+v2, v2+v3) What is the preimage of w=(0,2,5) ? I tried setting up the system of equations and got v1+v2= 2 and v2+v3=5 but after that I got kinda lost in how to find the individual solutions?