Let T Find the image of an arbitrary vector V₁ Let T: R² R² be the linear transformation satisfying T -30 23 r()=[ ] [] ([;)) = = and ₂ and = [2]. 21 T(₂2) = [248] -18

please write answer as a formula that returns a vector

| Let T : R? → R? be a linear transformation that sends the vector u = (5, 2) into (2, 1) and maps v = (1, 3) into (-1,3). Use properties of a linear transformation to calculate the following. (Enter your answers as ordered pairs, such as (1,2), including the parentheses.) T(3u) = T(-83) = T(3u – 80)

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just tell me linear or not linear plz

(2)

Suppose T: R2-→R³ is a linear transformation. Let U and V be the vectors given below, and suppose that T(U) and T(V) are as given. Find T(-3U-2V). 14 18 U = V = T(U) = -7 T(V) =-9 3 16 21 T(-3U–2V) = | 0

Let denote the vector space of polynomials in the variable x of degree n or less with real coefficients. Let D: 03 → be the function that sends a polynomial to its derivative. That is, D(p(x)) = p'(x) for all polynomials p(x) E 3. Is D a linear transformation? Let p(x) = a3x³ + a₂x² + a₁x + aº and q(x) = b3x³ + b₂x² + b₁x + bo be any two polynomials in 3 and c E R. a. D(p(x) + q(x)) = D(p(x)) + D(q(x)) = Does D(p(x) + q(x)) = D(p(x)) + D(q(x)) for all p(x), q(x) = ? choose b. D(cp(x)) = c(D(p(x))) = Does D(cp(x)) = c(D(p(x))) for all c ER and all p(x) E 3? choose c. Is D a linear transformation? choose . (Enter a3 as a3, etc.)

Consider a constant vector B e R*. Is the (projection) function F(ā)=| a-B à - dot products – where äeR“ a linear transformation? Justify, always.