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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q22E (page 164)

If A is an invertible n 脳n matrix, then the kernels of A and $A^{- 1}$ must be equal.

Short Answer

Expert verified

The above statement is true.

If A is an invertible n 脳n matrix, then the kernels of A and $A^{- 1}$ must be equal.

Step by step solution

01

Definition of kernel of a matrix

Let A be a matrix of order m 脳n, then the kernel of the matrix A, denoted by ker(A), is defined as-

$K e r (A) = \{x 鈭� V : A x = 0\}$

02

To determine the kernel of a matrix A and A-1

Since A is an invertible n 脳n matrix, then the rank of A and $A^{- 1}$ is 鈥榥鈥�.

So, the nullity of both matrices A and $A^{- 1}$ is zero.

This implies that,

$D i m (\ker (A)) = D i m (\ker (A^{- 1})) = 0$

$鈬� \ker (A) = \overset{鈫�}{0}, \ker (A^{- 1}) = \overset{鈫�}{0}$

$鈬� \ker (A) = \ker (A^{- 1}) = \overset{鈫�}{0}$

03

Final Answer

If A is an invertible n 脳n matrix, then

$D i m (\ker (A)) = D i m (\ker (A^{- 1})) = 0$

$鈬� \ker (A) = \ker (A^{- 1}) = \overset{鈫�}{0}$

Hence, if A is an invertible n 脳n matrix, then the kernels of A and $A^{- 1}$ must be equal.

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Most popular questions from this chapter

Find a basis of the subspace of ${鈩�}^{3}$ defined by the equation $2 x_{1} + 3 x_{2} + x_{3} = 0 .$

In Exercise 44 through 61, consider the problem of fitting a conic through $m$ given points $P_{1} (x_{1}, y_{1}), . ......, P_{m} (x_{m}, y_{m})$ in the plane. A conic is a curve in ${鈩�}^{2}$ that can be described by an equation of the form , $f (x, y) = c_{1} + c_{2} x + c_{3} y + c_{4} x^{2} + c_{5} x y + c_{6} y^{2} + c_{7} x^{3} + c_{8} x^{2} y + c_{9} x y^{2} + c_{10} y^{3} = 0$ where at least one of the coefficients $c_{i}$ is non zero. If is any nonzero constant, then the equations $f (x, y) = 0$ and $k f (x, y) = 0$ define the same cubic.

44. Show that the cubic through the points $(0, 0), (1, 0), (2, 0), (0, 1), and (0, 2)$ can be described by equations of the form $c_{5} x y + c_{7} (x^{3} 鈭� 3 x^{2} + 2 x) + c_{8} x^{2} y + c_{9} x y^{2} + c_{10} (y^{3} 鈭� 3 y^{2} + 2 y) = 0$ , where at least one of the coefficients is nonzero. Alternatively, this equation can be written as . $c_{7} x (x 鈭� 1) (x 鈭� 2) + c_{10} y (y 鈭� 1) (y 鈭� 2) + x y (c_{5} + c_{8} x + c_{9} y) = 0$

Determine whether the following vectors form a basis of ${鈩�}^{4}$ ; $[\begin{matrix} 1 \\ 1 \\ 1 \\ 1 \end{matrix}], [\begin{matrix} 1 \\ - 1 \\ 1 \\ - 1 \end{matrix}], [\begin{matrix} 1 \\ 2 \\ 4 \\ 8 \end{matrix}], [\begin{matrix} 1 \\ - 2 \\ 4 \\ - 8 \end{matrix}]$ .

Give an example of a parametrization of the ellipse

$x^{2} + \frac{y^{2}}{4} = 1$

in ${鈩�}^{2}$ . See Example .

Give an example of a linear transformation whose image is the line spanned by $[\begin{matrix} 7 \\ 6 \\ 5 \end{matrix}]$ in ${鈩�}^{3}$ .

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