Let T: R → R such that ker T = {(2s, s, t, s+t, 3t) : s,t e R}, then one of the following is true %3D A) T is onto but not one - to - one B) T is isomorphism C) T is one - to - one but not onto D) No such linear map exists

Q: Question 2. Let T : R2 → R³ and S : R³ –→R be linear maps. Are the following compositions defined?…

A:

Q: Let Q be the parallelogram in R2 whose vertices are (1, 1), (3, 2), (4, 5), (2, 4). Find the map T…

A: As per the question we first have to find the linear transformation T, whose inverse which will…

Q: Which of the following statements are true? I: A linear transformation from R to R" is called a…

A: The convex hull  of a shape is the smallest convex set that contains it.

Q: Linear operator T: R2 R2 is defined by T(1,0) (1,4), T(1,1)-(2,2) T one-to-one?

A:

Q: a) Define a mapping T:R* → R³ by [2a-b] =b+c T c+d i) Determine whether T'is a linear…

A:

Q: b) Is there a linear map T: V → W, where dim V = 3, dim W = 4, rank T = 2, and dim ker T = 1?

A:

Q: 3: Show that if T : R³ →R is a non-zero linear map, then it must be given by T = ax + by + cz for…

A:

Q: Let T : R3 −→ R3 be a linear map such that T () =, T () =, T () = . Compute T ().

A:

Q: If the range of the linear transformation T :R° → R' is R³ then the nullity of T is a) 2 b) 3 c) 4…

A: Rank-Nullity theorem: Let  T : V→W be a linear transformation, with V a finite-dimensional vector…

Q: 4: Let T : R? → R? be the transformation mapping defined by T(u, v) = (u + 1, v + 2) for all (u, v)…

A: Given that T:ℝ2→ℝ2 is the transformation mapping defined by ​T(u, v)=(u+1, v+2) for all (u,…

Q: Show that dim(W1) + dim(W2) – dim(W1 n W2) = dim(W1 + W2), where W1 + W2 just denotes the span of W1…

A:

Q: Which of the following maps from R³ to R³ are not linear? U □ f(x, y, z)=(x, y, z + 1). f(x, y, z) =…

A:

Q: Let X = {a,b,c, d},T= {0,X, {d}, (a,b}, {a,b, d}} and Y= {m,n,P. q},F = {¢,Y, {p}, {m,n}, {m,n,p}}…

A:

Q: Show that rectangle CAMP and rectangle C*A*MP are similarby identifying the composition of…

A: The given data is:

Q: 1. Let V = F∞ and consider the two linear maps T, S = L(F∞, F∞ defined by T(a₁, A2, A3, .) = (a2,…

A: Let  and consider  the  two linear  maps defined  by…

Q: 2. Which one(s) of the following mappings is(are) a linear mapping? Give the matrix of it(them).…

A:

Q: Let V, W, Z be vector spaces and let T:V-->W and U:W-->Z be linear transformations. Show that if T…

A:

Q: Let T : P4 P4 be linear map such that T(p) = p", then rank(T) = %3D A) 4 В) 3 С) 2 D) 5

A:

Q: 3) Given the mapping T: R² → R³ defined by T(x1 , X2) = (3x1 – 2.x2, 4.xj + 5x2, x1 –x2) a) Prove…

A:

Q: b) S : M3,3 → M3,3 defined by S(A) = –A"? (i.e. negative transpose)

A: This is a question of linear transformation. The problem is to verify whether the transformation is…

Q: Let L:R' →R is linear mapping. Knowing that L(1, 3,1) = 4,L (0,1, 2)=3,L(2, - 1,0)=1 Find a) L(1,…

A:

Q: (a) Write T as multiplication by a constant matı b) Solve the equation T (c) Solve the equation T

A:

Q: Let 7,V be a projection af the Vect or space. Accordling to this: Show that every Xe Im (T) for…

A: The given problem is related with projection. Given that T is a projection of the vector space V .…

Q: Let T: R2 R be a linear map such that T(x, y) = (2x, 3y, 2y- r), then one of the following in range…

A:

Q: 5. Let T: R2 → R² be a linear transformation such that Nullspace (T) = {0}. Show that T takes: a)…

A: let T:ℝ2→ℝ2 be a linear transformation such that nullspaceT=0 to show that T takes a) parallel lines…

Q: Consider the map T: R³ → R2 given by Determine the Image of T m(T) = span {(0,3),(1,0)} Im(T) =…

A: The map T:ℝ3→ℝ2 given by Tx1, x2, x3=x2-2x3, 3x1+x3 To find the image of T. Ifv1, v2, ...vn in the…

Q: -1 2 2 2 3 2 -4 0 A = 1. Define a transformation T: x Ⓡ Ax, where a) Find the image of u under T; b)…

A:

Q: Let V and W be finite-dimensional inner product spaces. Let T: →W and U: W →V be linear…

A:

Q: Let T, be the orthogonal projection onto the z axis, T, be the rotation with respect to x axis by 0.…

A:

Q: Let f: RnRm be a linear map that preserves linear independence, i.e. if (v₁,..., Vk) is a family of…

A:

Q: Define the following a) span {v₁, v₂,v}, b) {v₁, v₂,vis linearly n dent, mation. m c) span {v₁, v₂,…

A:

Q: State which of the following projection functions from R? to R belong to L(IR?, R). Id1, Idı + Id2,…

A:

Q: . Show that dim(w,) + dim(W,)– dim(w, n w,) = dim(w, + w, - 1 C

A:

Q: Let E and F be two vector spaces with the same finite dimension dim(E) = dim(F) For every linear map…

A: Follow the steps.

Q: 3. Let T: M₂2 → R² be defined as T ( [])=(a - b, cd). Determine whether or not I' is a linear…

A: I have given the answer in the next step. Hope you understand that

Q: Suppose T: V --> W is a linear and surjective map and V and W are of finite dimension. Show that T…

A:

Q: Suppose b, c E R. Define a map T : R³ → R² by T(x, y, z) = (2x − 3y + b, 4y — cxz). Show that T is…

A: