Find out which of the transformations in Exercises 1 through 50 are linear. For those that are linear, determine whether they are isomorphisms.
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Linear Algebra With Applications
- Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).arrow_forwardLet T be a linear transformation from P2 into P2 such that T(1)=x,T(x)=1+xandT(x2)=1+x+x2. Find T(26x+x2).arrow_forwardFor the linear transformation T:R2R2 given by A=[abba] find a and b such that T(12,5)=(13,0).arrow_forward
- Let T:RnRm be the linear transformation defined by T(v)=Av, where A=[30100302]. Find the dimensions of Rn and Rm.arrow_forward.Define f: R- R by f(x) = rx + d, where r and d are real constants. Show that for f to be linear, it is required that d 0. (the definition of a linear transformation is given in text section 1.8) ()- 4r1-2 . Is the transformation T linear? YES or NO (circle one) I2 X122+3 Justify your answer:arrow_forwardWhich of the following transformations are linear? Select all of the linear transformations. There may be more than one correct answer. Be sure you can justify your answers. □A. T(f(t)) = (f(t))³ + 6(f(t))² + 5f (t) from C to C B. T(x0, X₁, X2, ...) = (1, X0, X₁, X₂, ...) from the space of infinite sequences into itself C. T(f(t)) = f(9) from P7 to R D. T(f(t)) = f(-t) from P5 to P5 □E. T(f(t)) = f'(t) + 6f (t) from C* to C* = [₁₁ □ F. T(f(t)) = f(t)dt from P3 to Rarrow_forward
- II. Let V =span- and define the transformations [a b] c d [a² – 6? c? - d²1 S(A) = adj(A) d Determine which transformations are linear. i. T: M2x2(R) → M2x2(R) ii. S: M2x2(R) → M2x2(R) iii. T :V → M2x2(R)arrow_forwardSuppose a transformation has a mapping as follows:(x,y)→(1/2 x + 3, -2y + 4). If this mapping were applied to the function y = |x|, write the equation of the transformed function in the form y = af [k (x - d) ] + c.arrow_forwardDetermine if the statements below are True or False.If it’s True, explain why. If it’s False explain why not, or simply give an exampledemonstrating why it’s false If T : R3 → R3is a function which has the property T(ax + by) = aT(x) + bT(y) forall a, b ∈ R and all x, y ∈ R3, then T is a linear transformation.arrow_forward
- I know that a) is not linear and b) and c) are linear so could you answer it for d) and e) and f) please?arrow_forwardPlease answer both a and b with explanation Thanksarrow_forwardLet T : P2 → P2 be a function represented by T(c+ bx + ax?) = (3c+ a) + (2b + 3a)x + ax? Determine whether T is a linear transformation.arrow_forward
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