Linear transformation Definition and 437 Threads

Let the set S be a set of linearly independent vectors in V, and let T be a linear transformation from V into V. Prove that the set {T(v_1), T(v_2),...,T(v_n)} is linearly independent. We know that any linear combination of the vectors in S, set equal to zero, has only the trivial solution...

how would one find the inverse of the linear transformation: y_1=4x_1-5x_2 y_2=-3x_1+4x_2 this was never taught in class, could someone give a little advice as how I would do this? I know the answer has to be in the form of x_1=ay_1+by_2 x_2=cy_1+dy_2 could someone explain this...

This is how the question appears in my textbook Find the matrix of T corresponding to the bases B and D and use it to compute C_{D}[T(v)] and hence T(v) T; P2 - > R2 T(a + bx + cx^2) = (a+b,c) B={1,x,x^2} D={(1,-1),(1,1)} v = a + bx + cx^2 ok i cna find Cd no problem it is C_{D}[T(v)]...

Find a basis for Ker T and a basis for I am T a) T: P_{2} -> R^2 \ T(a+bx+cx^2) = (a,b) for Ker T , both a and b must be zero, but c can be anything so the basis is x^2 for hte image we have to find the find v in P2 st T(v) = (a,b) \in P^2 the c can be anything, right? cant our basis be...

Find a linear transformation with the given properties T(2,-1) = (1,-1,1) and T(1,1) = (0,1,0) we need to find an expression for t(x,y) so we could find what linear combo on (2,-1) and (1,1) yierlds x,y But i tried that and i find that i cannot solve this system of linear equations with...

Indicate whether each statement is always true, sometimes true, or always false. IF T: R^n --> R^m is a linear transformation and m > n, then T is 1-1 Not sure to how prove this..

find the inverse of T \left[ \begin{array}{cc} a&b \\ c&d \end{array} \right] = \left[ \begin{array}{cc} a+2c&b+2d \\ 3c-a&3d-b \end{array} \right] do i row reduce the transformation matrix... it doesn work , though is there an easier way??

Let g(x)\in F[x], T\in L(V). Let F[T] be a ring generated by g(T). Show that if g(T) is invertible, then g^{-1}(T)\in F[T]. No idea what do do.

Hello everyone, I'm studying an example my professor did, and it isn't making sense to me... here is the orignal matrix: THe oringal matrix is: T[s] = [3s-t] [t]...[2t+7s] he wants to determine if the following trnasformation is Linear. Here is what he wrote on the board...

Hello everyone, I'm confused on this problem: It says: A linear transformation T:R^3->R^2 whose matrix is 2 -4 -3 -3 6 0+k is onto if k != ? != meaning, not equal. So my thinking was, For it to be a transformation into R^2, doesn't that mean k isn't suppose mean that the column -3...

I saw a similar post to this one, but i just got lost in the mess of the whole thing. So i just started a new thread. A question reads: Let T: Pn ---> Pn be defined by T[P(x)] = p(x) + xp'(x), where p'(x) denotes the derivative. Show that T is an isomorphism by finding Mbb(T) when B =...

Hi, suppose I have a linear transformation T and Ker(T) consists of only the zero vector. Then is it true that a basis for Ker(T) consists of no vectors and is of dimension zero? I would like these technicalities to be clarified. Any help would be good thanks.

Linear transformation - matrices *edit* Question resolved (y) I'm not sure what to do in the following question. A linear transformation has matrix P = \left[ T \right]_B = \left[ {\begin{array}{*{20}c} 3 & { - 4} \\ 1 & { - 1} \\ \end{array}} \right] with respect to the standard basis...

Okay, I will just admit that I stink at using mathematical proof in Linear. I hope someone can give me a push with this problem Prove that T : R(real)^3 -> R(real)^3 defined by T([yz,xz,zy]) is not a linear transformation. Reading my book I know that I need to prove that the transformation...

I have a question regarding a math problem that I do not know how to go about solving. Let L: R^2 ---> R^2 be a linear operator. If L((1,2)^T)) = (-2,3)T and L((-1,1)) = (5,2)^T determine the value of L((7,5))^T Any insight would be much appreciated.

check work please on linear transformation problem The problem is to find a standard matrix of T. T:\mathbb{R}^3\rightarrow\mathbb{R}^2, T(\vec{e}_1) = (1,3), T(\vec{e}_2) = (4,-7), T(\vec{e}_3) = (-5,4) where e_1, e_2, and e_3 are the columns of the 3x3 identity matrix. So here's what I did...

Let T:R^3 -> R be linear. Show that there exist scalars a, b, and c such that T(x, y , z) = ax + by + cz for all (x, y, z) in R^3. State and prove an analogous result for T: F^n -> F^m. I know that we just have to multiply by a matrix then we can get the desired transformation. But how would...

This is probably a simple question, but just to be sure: if the kernel of linear transformation is {0}, then the set is linearly dependent since 0-vector is LD, right? So dimension is 0, right? Then what's the basis of kernel? No basis? thanks in advance.

Hi, can someone please check my working for the first part and help me out with the second bit? Q. Let P_2 be the vector space of all polynomials of degree at most 2. A function T:P_2 \to R^3 is defined by T\left( {p\left( x \right)} \right) = \left( {p(1),p'(1),p''(1)} \right). That is, T...

Hi Guys, I have these linear transformation problems which have caused me some trouble today. I hope You can help me. a) (x,y) \rightarrow (x+3,y+5) is called a linear translation according to my Linear Algebra textbook. I'm tasked with showing that the above can't be done as...

Hi, I have that |T(p)| <= sqrt(10)*|p| where T is a linear mapping. The question is: How small must |p' - p''| be in order that |T(p') - T(p'')| <= 1/10. This is what I did: T linear, so |T(p') - T(p'')| = |T(p' - p'')|. Applying the bound: |T(p' - p'')| <=...

\ Let T: V \rightarrow W be a linear transformation, let b \in W be a fixed vector, and let x_0 \in V be a fixed solution of T(x)=b. Prove that a vector x_1 \in V is a solution of T(x)=b, if and only if x_1 is of the form x_1=x_h +x_0 where x_h \in kerT I started out by saying that...

Hi Ho! ^_^ I stuck when doing David C. Lay's Linear Algebra in Exercise 1.8 about Linear Transformation I'm asked to determine whether these statements are correct. Statement 1: A linear transformation is a special type of function. Statement 2: The superposition principle is a physical...

OK I already have the answer for this problem but I don't know how my teacher came up with the answer: Linear transformation T in R^3 consists of the rotation around x3 axis at the positive (counter-clockwise) direction at the angle 90 degrees. Such rotation transforms x1-axis into x2-axis...

Hi Ho! ^^v I've some questions regarding linear transformation in my linear algebra course, guys! Please help me! ^^v Statement: A linear transformation is a special type of function. My answer: Yes, it is a special type of function because it must satisfy the following properties from...

Hi I have a linear transformation T which maps \mathbb{R}^3 \rightarrow \mathbb{R}^2 a A is the standard matrix for the linear transformation. I'm suppose to determain that T maps \mathbb{R}^3 \rightarrow \mathbb{R}^2 I was told by my professor about the following theorem. If...

Hi I got a question regarding the matrix of linear transformation. A linear transformation L which maps \mathbb{R}^{3} \rightarrow \mathbb{R}^2 implies that L(2,-1,-1) = (0,0) and L(-1,2,1) = (1,3) and L(2,2,1) = (4,9). My question is: The matrix of linear transformation is that then...

I have a linear map from $ V\rightarrow K[X_{1},...,X_{n}]\rightarrow K[X_{1},...,X_{n}]/I.$ how do i prove that a linear map from $ V=\{$polynomials with $\deg _{x_{i}}f\prec q\}$ to $ K[X_{1},..X_{n}]/I.$ where I is the ideal generated by the elements $ X_{i}^{q}-X_{i},1\leq i\leq n.,$ is...

Hello all, I am trying to understand the matrix representation of a linear transformation. So here is my thought process. Let B = (b1, b2, ..., bn) be a basis for V, and let Y = (y1, y2, ..., ym) be a basis for W. T: V --> W Pick and v in V and express as a linear combo of the...

i was trying to figure out something that i didn't understand and the book doesn't have much examples of it either. My question is how do u know whether a transformation is a projection on a line, reflection on a line, or rotation through an angel? With T given. The questions i did from the...

Let f: R --> R and let T: P2 --> F, and T(p) = p(f). Prove that T is a linear transformation. P2 is the set of polynomials of degree 2 or less, and F is the set of all functions. It seems to me that I can treat f as really just a real number, in which case it's no different from proving...

take a point A(x_1,y_1) on a circe centre B (x_2, y_2) and allow the circle to roll along an X-axis; now we all know that the cycloid equation to point A is highly non-linear; so now if we take the point B, we find the problem has been converted into a linear problem; now do this to E-fields...

hi, Could someone please show me how these are a linear transformation please: 1) T(s u + r V) s and r are scalars and u and v are vectors. 2) composite function: u : v thanks

Linear Transformation -- Onto I'm having trouble with the first part of the following problem: Let T be a linear transformation from an n-dimensional space V into an m-dimensional space W. a) If m>n, show that T cannot be a mapping from V onto W. b) if m<n, show that T cannot be...

Linear transformation in Maths (a) If a triangle ABC with coordiantes A(2,7), B(2,9) and C(6,7) has a rotation and maps to triangle PQR with coordinates P(6,5), Q(8,5) and R(6,1), what is the centre of rotation? I want to ask in general, what's the way to find the answer? (b) An enlargement...

(a) If a triangle ABC with coordiantes A(2,7), B(2,9) and C(6,7) has a rotation and maps to triangle PQR with coordinates P(6,5), Q(8,5) and R(6,1), what is the centre of rotation? I want to ask in general, what's the way to find the answer? (b) An enlargement maps the triangle ABC with...

How will I prove that... Show that L: V -> W is a linear transformation if and only if L(au + bv) = aL(u) + bL(v) for any scalars a and b and and any vectors u and v in V. For L(au +bv), this is my proof. (Is this wrong?) L(au + bv) = L [ a(a', b', c') + b(a'', b'', c'')]...