If T : V → W and U : W → Z are linear transformations, prove that rank(UT) < rank(T). (Hint: How does N(UT) compare with N(T)?]

Q: Consider the map T : M,(R) → R defined by T(A) = det(A). Is T a linear transformation? If yes, prove…

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Q: Let f : R? → R be defined by f((x, y)) = 5x + 7y. Is ƒ a linear transformation? c. Is f a linear…

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Q: In Question, determine whether T is a linear transformation.

A: Given: T(x)=yxTy. Determine T(x1+x2). T(x1+x2)=yx1+x2Ty=yx1Ty+yx2Ty=T(x1)+T(x2)

Q: Find the kernel of the linear transformation.T: R3→R3, T(x, y, z) = (−z, −y, −x)

A: Here the given linear transformation Use the definition kernel of the linear transformation

Q: Show that the function D : Vn(I) → Vn(I) defined by D(x(t))=x′(t) is a linear transformation.

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Q: Suppose T is the transformation from ℝ2 to ℝ2 that results from a reflection over the y-axis…

A: Given, T is the transformation from ℝ2 to ℝ2 that results from a reflection over the y-axis followed…

Q: Let (x, y, z) E R³ and the transformation T: R³ → R² be given by T(x, y, z) = (2x + 4y, x + 3y + z).

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Q: Suppose T is the transformation from R- to R- that results from a reflection over the line y=-x…

A: Given query is to find the matrix A which induces T.

Q: (9) Let T: R. -R? be a linear transformation. Suppose T(u) = and T(v) = What is T(2u -3v)?

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Q: 25) Show that the linear transformation T(x, y, z) = (x - 2y, -x+y+z, x+y) is n isomorphism, and…

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Q: Why is the question “Is the linear transformation T onto?" an existence question?

A: A linear transformation T:Rn -&gt; Rm is onto, if for every vector w in Rm, there exist an vector v…

Q: The linear transformation T(f) = f+ f'' from C 0 C is an isomorphism. True False

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Q: Let C1,2 be the vector space of continuous functions mapping the interval 1 <x<2 into R. Let T:…

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Q: prove that the given transformation is a linear transformation, using the definition

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Q: ) Let T : R² → R³ be a linear transformation such that 2 and T(( 1 (:) Is the vector u = (*) in R…

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Q: en 9. Ift:V→Vis any map from an inner-product space V to itself such that (i) t (0) = 0 (ii) || t…

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Q: 2. The mapping T: R' R defined as T(u) = ||u|| is NOT a linear transformation. OTrue False

A: the mapping T ; R 2 - R defined as T (w) = u is not a linear transformation .  we have to tell it…

Q: prove that the given transformation is a linear transformation, using the definition

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Q: Give a counterexample to show that the given transformation is not a linear transformation.

A: Given         T xy = xyx+y

Q: give a counterexample to show that the given transformation is not a linear transformation.

A: Given,   A linear transformation,

Q: Use the lemma in this section to show that if T is an invertible linear transformation then ||T|| >…

A: ?Let , T be a linear Transformation then since T  is invertible linear Transformation. Therefore ,…

Q: If T: R3 → R is a linear transformation such that T (E) then T

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Q: Let T1 : R" → R* and T2 : R* → R" be linear transformations. Prove that for all k e R, and ve R":…

A: Idea: using definition of linear transformation  T(kv)=kT(v)

Q: Let T1, T2 : V → V be linear transformations satisfying Tị • T2 = ly (where 1y denotes the identity…

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Q: Find the kernel of the linear transformation.T: R3→R3, T(x, y, z) = (x, 0, z)

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Q: Let T : R3 R' be the linear transformation defined by Т(х, х2, хз) — (х1 — X2, Xә — Хз, Xз — X).…

A: A transformation is a general term for four specific ways to manipulate the shape and the position…

Q: Let (x, y, z) E R and the transformation T: R³ → R² be given by T(x, y, 2) = (2.x + 4y, x + 3y + 2).

A: Transformation is invertible if and only if transformation is square.

Q: Is the transformation below linear? If so, is it also an isomorphism? T(f(t)) f(t)dt from P2 to R…

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Q: Show that the function T : Pn → R defined by T(r0+r1x+· · ·+rnx^n) = rn is a linear transformation.

A: Given that, the functionT:Pn→R defined byT(r0+r1x+···+rnxn)=rn

Q: Let F: R3 → R4 be a linear transformation.   f (x, y, z) = (x, x + 3y, x, x + z) Is f bijective?

A: A function f is said to be bijective if and only if it is both injective(one to one) and surjective…

Q: If T : R –→ R° is a linear transformation such that (E) 2 1 T T 2 T 3 3 --) (E) then T || ||

A: Let's find.

Q: Prove that the given transformation is a linear transformation,

A: Txyz=x+zy+zx+y

Q: For the linear transformation T(r,y,z)= (r-z,y+z,x+y) ker(T)= nullity(T)= : and rank(T)=

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Q: The function T: M22→R defined by T(A)=det(A) is not linear transformation. True False

A: If A is the matrix of order 2 by 2 such that the transformation: T(A)=det(A) The transformation T is…

Q: 7. Show that the transformation T: R →R defined by T (x, y, z) = (x– y, 0, y + z) is a linear…

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Q: Let L : R* → R“ be a linear transformation defined by X3 x2 The range of L consists of all the…

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Q: give a counterexample to show that the given transformation is not a linear transformation.

A: Given,  The transformation

Q: Find the kernel and nullity of the transformation T. T(f(t)) f(t)dt from P, to R

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Q: 3. Let the transformation T : R² → R³ via T(x, y) = (√√, xy, √7). Determine whether T is a linear…

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Q: Which of the following is NOT a linear transformation? (E) (E) L: R° → Rdefined by L = 0 L: R3 →…

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Q: Suppose T is the transformation from R2 to R2 that results from a reflection over the line y=-x…

A: The solution are next step

Q: Let T: R3--&gt;R4 be a linear transformation such that the only solution to T(x)=0 is the trivial…

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Q: Let T1, T2 : V → V be linear transformations satisfying Tị • T2 = ly (where 1y denotes the identity…

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Q: Let T:R + R be the linear transformation defined by - (E)-H - () -- -()-A T T Is this linear…

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Q: Let T: R7 → M₂2 be a linear transformation. If the dimension of the kernel of T is 4, what is…

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Q: If a transformation T: R" → Rm satisfies T(0) = 0, then it is a linear transformation. True False

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Q: Decide whether the following statement is true or false. If a linear transformation T R" R" is…

A: T is linear transformation then T inverse also linear transformation. since T is invert able.

Q: If L :V → W is a linear transformation which of the following is FALSE?

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Q: Suppose T: R² → R²is defined by T(x , y) = (x + y ,x ) then Tis .a Linear transformation .b…

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