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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 2: Q 43E (page 98)

Question:Consider two invertible n 脳 n matrices Aand B. Is the linear transformation $\overset{鈫�}{y} = A (B \overset{鈫�}{x})$ invertible? If so, what is the inverse? Hint: Solve the equation $\overset{鈫�}{y} = A (B \overset{鈫�}{x})$ first for $B \overset{鈫�}{x}$ and then for $\overset{鈫�}{x}$ .

Short Answer

Expert verified

The linear transformation $\overset{鈫�}{y} = A (B \overset{鈫�}{x})$ is invertible and the inverse is $\overset{鈫�}{x} = B^{鈭� 1} (A^{鈭� 1} \overset{鈫�}{y})$

Step by step solution

01

Consider the system of matrices

Consider the system.

$\overset{鈫�}{y} = A (B \overset{鈫�}{x})$

The matrix is said to be invertible if and only if the determinant of the matrix is nonzero.

02

Compute the inverse

Consider the system,

$\overset{鈫�}{y} = A (B \overset{鈫�}{x})$

The inverse is,

$\begin{matrix} A^{鈭� 1} (\overset{鈫�}{y}) = A^{鈭� 1} [A (B \overset{鈫�}{x})] \\ A^{鈭� 1} \overset{鈫�}{y} = (A^{鈭� 1} A) (B \overset{鈫�}{x}) \\ A^{鈭� 1} \overset{鈫�}{y} = B \overset{鈫�}{x} \\ 鈭� \overset{鈫�}{x} = B^{鈭� 1} (A^{鈭� 1} \overset{鈫�}{y}) \end{matrix}$

Hence, the linear transformation $\overset{鈫�}{y} = A (B \overset{鈫�}{x})$ is invertible and the inverse is $\overset{鈫�}{x} = B^{鈭� 1} (A^{鈭� 1} \overset{鈫�}{y})$

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Most popular questions from this chapter

Consider the circular face in the accompanying figure. For each of the matrices A in Exercises 24 through 30, draw a sketch showing the effect of the linear transformation $T \overset{鈫�}{(x)} = A \overset{鈫�}{x}$ on this face.

30. $[\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}]$

TRUE OR FALSE?

If $A^{2} = I_{n}$ , then matrixAmust be invertible.

TRUE OR FALSE?

The matrix $(\begin{matrix} 1 & 1 \\ 1 & - 1 \end{matrix})$ represents a reflection about a line.

Consider the circular face in the accompanying figure. For each of the matrices A in Exercises 24 through 30, draw a sketch showing the effect of the linear transformation $T \overset{鈫�}{(x)} = A \overset{鈫�}{x}$ on this face.

26. $[\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}]$

Use the concept of a linear transformation in terms of the formula $y = A \overset{鈬赌}{x}$ , and interpret simple linear transformations geometrically. Find the inverse of a linear transformation from localid="1659964769815" ${鈩�}^{2} t o {鈩�}^{2}$ to (if it exists). Find the matrix of a linear transformation column by column.

Consider the transformations from ${鈩�}^{3} t o {鈩�}^{3}$ defined in Exercises 1 through 3. Which of these transformations are linear?

$y_{1} = 2 x_{2} y_{2} = x_{2} + 2 y_{3} = 2 x_{2}$

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