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Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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Question
Let T : R2[x] → R2[x] be given by T(a+bx+cx2 ) = a + b(x+ 1) + b(x + 1)2 . Let v = x + 1. Find a basis for <T, v>.
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- find a basis for the range from r2 to r3 as defined by [3x1+4x2; x1-2x2; 4x1]arrow_forwardConsider the operator Son the vector space by S(a+bx) = - a+b+ ( a + 2b) x A) Pick a basis B = { b₁,b₂3. Find the minimal polynomials NT, b, (X), Nr, 0₂ (x), and Ns (x) R₁ [x] given B) Show that S is cyclic by finding a vector v such thatarrow_forwardShow that T(x, y, z) = (4x + 2y – 2z, −2x + y + 3z, x - y - 2z) is not a one-to-one transformation from R³ to R³. Find a basis of the kernel of this transformation.arrow_forward
- Let S {V1, V2, V3} and T = {w₁, W2, W3} be two bases in the vector space P₂ {a+bx+cx²: a, b, c = R}, where V₁ = 1+x+3x², v₂ : −2 − 2x − 7x², V3 = 1 + 2x + 6x² and = : 1 + x − x², w₂ = W1 = 1+ 2x², W3 = 3-x+5x2 Find the transition matrix from T to S. Let v = 201 - 3v2 + V3, find the coefficients (C₁, C2, C3) so that v = C₁w₁ + C₂W2 + C3W3. Hint: For the second part, you need also to find the transition matrix from S to T. =arrow_forward3. Let F be a subfield of complex numbers. Let V be the vector space (over F), consisting of all polynomials, i.e. V = F[r]. Let T:V → V be the function defined by T: f(x) + (x + 1)f'(x). d For example T(x² + x) = (x + 1) . — (x² + x) = (x + 1)(2x + 1) = 2x² + 3x +1. dr (a) Is T a linear transformation? Give proper reasons. (b) Is T invertible? Give proper reasons.arrow_forward'arrow_forward
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