Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Consider the function $T (A) (\overset{⇀}{X}) = {\overset{⇀}{x}}^{T} A \overset{⇀}{x} from R^{nxn} to Q_{n}$ . Show that T is a linear transformation. Find the image, kernel, rank, and nullity of T.

the solution is

$\ker (T) = \{A \in R^{nxn} : A^{T} = - A\} img (T) = \{x^{T} Ax \in Q_{n} : A \in R^{nxn}, A = A^{T}\} rank (T) = \frac{n (n + 1)}{2} nulity (T) = \frac{n (n - 1)}{2}$

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: Given information

$T (A) (\overset{⇀}{x}) = \overset{⇀ T^{}}{x} A \overset{⇀}{x}$

Step 2: Linear transformation and image of T and kernel of T

Consider $A, B \in R^{nxn} and α \in R,$ then

$T (αA + B) (x) = x^{T} (αA + B) x = x^{T} (αA) x + x^{T} Bx = {αx}^{T} Ax + x^{T} Bx = αT (A) (x) + T (B) (x)$

$For α = 1, we get T (A + B) = T (A) + T (B) and for B = 0, we get T (αA) = αT (A) . So, T : R^{NXN} \to Q_{n}, T (A) (x) = x^{T} Ax is a linear transformation . Image of T$

We know that for any symmetric matrix $A \in R^{nxn}$ times $n, x^{T} Ax$ , creates a quadratic form(unique), and that the image of T defines the entire space of quadratic form $Q_{n}$ described by the symmetric matrices.

$rank (T) = \dim (lmg (T)) = \frac{n (n + 1)}{2}$

Kernel of T: if A is a skew symmetric matrix, that is $A^{T} = - A$ then ,

$(x^{T} Ax) = x^{T} A^{T} x = - x^{T} Ax But, since x^{T} Ax is a real number, so {(x^{T} Ax)}^{T} = x^{T} Ax and so, we get$

$x^{T} Ax = - x^{T} Ax \Rightarrow x^{T} Ax = 0$ . So, the space of all skew-symmetric matrices is the subset of the kernel of T. By rank-nullity theorem, dimension of kernel of T= nullity of T ${dimR}^{nxn} = n^{2} and rank (T) = \frac{n (n + 1)}{2}$ As a result, because is the dimension of the space of skew-symmetric matrices, all skew-symmetric matrices make up the whole kernel of T.

Thus,

$\ker (T) = \{A \in R^{nxn} : A^{T} = - A\} lmg (T) = \{x^{T} Ax \in Q_{n} : A \in R^{nxn}, A = A^{T}\} = Q_{n}$

Step 3: Conclusion

$\ker (T) = \{A \in R^{nxn} : A^{T} = - A\} lmg (T) = \{x^{T} Ax \in Q_{n} : A \in R^{nxn}, A = A^{T}\} rank (T) = \frac{n (n + 1)}{2} nulity (T) = \frac{n (n + 1)}{2}$

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Linear Algebra With Applications

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Short Answer

Consider the function $T (A) (\overset{⇀}{X}) = {\overset{⇀}{x}}^{T} A \overset{⇀}{x} from R^{nxn} to Q_{n}$ . Show that T is a linear transformation. Find the image, kernel, rank, and nullity of T.

Step by Step Solution

Step 1: Given information

Step 2: Linear transformation and image of T and kernel of T

Step 3: Conclusion

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Linear Algebra With Applications

Answers without the blur.

Short Answer

Consider the function T(A)(X⇀)=x⇀TAx⇀ from Rnxn to Qn . Show that T is a linear transformation. Find the image, kernel, rank, and nullity of T.

Step by Step Solution

Step 1: Given information

Step 2: Linear transformation and image of T and kernel of T

Step 3: Conclusion

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Consider the function $T (A) (\overset{⇀}{X}) = {\overset{⇀}{x}}^{T} A \overset{⇀}{x} from R^{nxn} to Q_{n}$ . Show that T is a linear transformation. Find the image, kernel, rank, and nullity of T.