Q46E Consider the linear transformati... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 8: Q46E (page 402)

Consider the linear transformation $T (q (x_{1}, x_{2}, x_{3})) = q (x_{1}, x_{2}, x_{1}) from Q_{3} {toQ}_{2}$ . Find the image, kernel, rank, and nullity of this transformation.

Short Answer

Expert verified

the solution is

$\ker (T) = \{q (x_{1}, x_{2}, x_{3}) = {ax}_{1}^{2} + {cx}_{3}^{2} + {dx}_{1} x_{2} - {dx}_{2} x_{3} - (a + c) x_{3} x_{1} : a, c, d 鈭� R\} im (T) = Q_{2} rank (T) = 3 nullity (T) = 3$

Step by step solution

given information

$T (q (x_{1}, x_{2}, x_{3})) = q (x_{1}, x_{2}, x_{1})$

find kernel of T and image of T

$\ker (T) = \{q (x_{1}, x_{2}, x_{3}) 鈭� Q_{3} : T (q (x_{1}, x_{2}, x_{3})) = 0 鈭� Q_{2}\} = \{q (x_{1}, x_{2}, x_{3}) 鈭� Q_{3} : q (x_{1}, x_{2}, x_{1}) = 0 鈭� Q_{2}\} = \{q (x_{1}, x_{2}, x_{3}) = a x_{1}^{2} + b x_{2}^{2} + c x_{3}^{2} + d x_{1} x_{2} + e x_{2} x_{3} + f x_{3} x_{1} 鈭� Q_{3} : q (x_{1}, x_{2}, x_{1}) = 0\}$

$= {q (x_{1}, x_{2}, x_{3}) = {ax}_{1}^{2} + {bx}_{2}^{2} + {cx}_{3}^{2} + {dx}_{1} x_{2} + {ex}_{2} x_{3} + {fx}_{3} x_{1} 鈭� Q_{3} : {ax}_{1}^{2} + {bx}_{2}^{2} + {cx}_{1}^{2} + {dx}_{1} x_{2} + {ex}_{2} x_{1} + {fx}_{1}^{2} = 0}$

$= {q (x_{1}, x_{2}, x_{3}) = {ax}_{1}^{2} + {bx}_{2}^{2} + {cx}_{3}^{2} + {dx}_{1} x_{2} + {ex}_{2} x_{3} + {fx}_{3} x_{1} 鈭� Q_{3} : a + c + f = 0, b = 0, d + e = 0} = q (x_{1}, x_{2}, x_{3}) = {ax}_{1}^{2} + {cx}_{3}^{2} + {dx}_{1} x_{2} - {dx}_{2} x_{3} - (a + c) x_{3} x_{1} : a, c, d 鈭� R$

role="math" localid="1659673332347" $im (T) = \{T (q (x_{1}, x_{2}, x_{3})) 鈭� Q_{2} : q (x_{1}, x_{2}, x_{3}) 鈭� Q_{3}\} = q (x_{1}, x_{2}, x_{1}) 鈭� Q_{2} : q (x_{1}, x_{2}, x_{3}) = a x_{1}^{2} + b x_{2}^{2} + c x_{3}^{2} + d x_{1} x_{2} + e x_{2} x_{3} + f x_{3} x_{1} 鈭� Q_{3} = Q_{2}$

$rank (T) = \dim (im (T)) = 3 nullity (T) = \dim (\ker (T)) = 3$

conclusion

$\ker (T) = \{q (x_{1}, x_{2}, x_{3}) = {ax}_{1}^{2} + {cx}_{3}^{2} + {dx}_{1} x_{2} - {dx}_{2} x_{3} - (a + c) x_{3} x_{1} : a, c, d 鈭� R\} im (T) = Q_{2} rank (T) = 3 nullity (T) = 3$

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91影视

Consider the linear transformation $T (q (x_{1}, x_{2}, x_{3})) = q (x_{1}, x_{2}, x_{1}) from Q_{3} {toQ}_{2}$ . Find the image, kernel, rank, and nullity of this transformation.

Short Answer

Step by step solution

given information

find kernel of T and image of T

conclusion

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91影视

Consider the linear transformation T(q(x1,x2,x3))=q(x1,x2,x1)fromQ3toQ2. Find the image, kernel, rank, and nullity of this transformation.

Short Answer

Step by step solution

given information

find kernel of T and image of T

conclusion

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Probability and Statistics

Geometry

Logic and Functions

Decision Maths

Pure Maths

Study anywhere. Anytime. Across all devices.

Consider the linear transformation $T (q (x_{1}, x_{2}, x_{3})) = q (x_{1}, x_{2}, x_{1}) from Q_{3} {toQ}_{2}$ . Find the image, kernel, rank, and nullity of this transformation.