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Elementary Linear Algebra (MindTap Course List)

Elementary Linear Algebra (MindTap Course List)

8th Edition

ISBN:9781305658004

Author:Ron Larson

Publisher:Ron Larson

Chapter7: Eigenvalues And Eigenvectors

Section7.CM: Cumulative Review

Problem 16CM

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Let T be the linear transformation from the space of degree three or less polynomials
to itself defined by the following formula:
T(ax³ + bx² + cx + d) = (a − b)x³ + (a - b)x² + (2c+ d)x + (2c+ d)
T.
Find a basis for the image of

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Find a basis for R2 that includes the vector (2,2).
Let T be a linear transformation from R2 into R2 such that T(4,2)=(2,2) and T(3,3)=(3,3). Find T(7,2).
Find the kernel of the linear transformation T:R4R4, T(x1,x2,x3,x4)=(x1x2,x2x1,0,x3+x4).
Show two examples for the polynomials
Determine if the specified linear transformation is (a) one-to-one and (b) onto. Justify each answer. T(X1 X2 X3) = (x₁ - 4x2 +6x3, X2-9x3) (a) Is the linear transformation one-to-one? O A. T is one-to-one because the column vectors are not scalar multiples of each other. B. T is one-to-one because T(x) = 0 has only the trivial solution. O C. T is not one-to-one because the columns of the standard matrix A are linearly independent. O D. T is not one-to-one because the columns of the standard matrix A are linearly dependent.
Let T be the linear transformation from the space of degree three or less polynomials to itself defined by the following formula: T(ax³ + bx² + cx + d) = (a − b)x³ + (a - b)x² + (2c+ d)x + (2c+ d) T. Find a basis for the image of
,
Let T be a linear transformation from R³ into R³. Find T-1 T(x1x₂x3) = (2x1-X3, X₁ + X₂ X3, X2-3X3) T(X₁₁X₁₁X3) = (2x₁+x₂ −X3, −3x₁+6x₂−x3, −X₁+2×₂-2x3) b. T(x1x2x3) = (2x₁ + x₂ -2X3, X₂-X3, X1 + X3) c. T(x₁,x₂₁x3) = (2x₁ + x₂ -2X3, X₂-X3, -X₁ + X3) d. T(x1,x2x3) = (4x₁-2×₂-X3, X₁-X₂, −3x₁+2x₂ + x3) T(X₁, X₂2₁×3) = (-x₁ + 3x₂ + 3x3, −3x₁-x₂ + 4x3,2x₁ − ×2 −X3) a. e.
Let T be a linear transformation from R³ into R³. Find T-1 T(X1₂X2₂X3)=(X₁ + X3, X₁−X₂ + X3, X₁ + 2x₂ + 2x3) a. T(×₁,×2,×3)=—=—(2×₁+x₂−X3₁ −3×₁+6×₂-X3₁ −X₁+2x₂-2x3) b. T(x1x₂x3) = (2x₁ + x₂-2X3, X₂ X3, X1 + X3) X1 + X3) c. T(x1,x2x3) = (2x₁ + x₂ -2X3, X₂ X3 d. T(x₁,x₂,X3)= (4x₁-2×₂-X3, X₁-X₂, 3X₁ + 2x₂ + x3) T(X1₁X₁₁X3)= (-X₁+3x₂ + 3x3, −3x₁-x₂ + 4x3,2x₁ −x₂ −X3)
Determine whether or not the following linear transformation is invertible, and justify your answer
Determine whether the function L: P2 → P2 defined by L(a0 +a1x+a2x2) = a1 +2a2xis a linear transformation. Explain, and show all your work.
Suppose T: P3→R³ is a linear transformation whose action is defined by a-2b+5c-4d T(ax³ + bx²+cx+d) = 3a-b+5c-2d a-b+3c-2d Determine whether T is one-to-one and/or onto. If it is not one-to-one, show this by providing two polynomials that have the sam the image of T. Use the '^' character to indicate an exponent and x as the variable for polynomials, e.g. 5x^2-2x+1 T is one-to-one T is onto

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