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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: Q59E (page 109)

TRUE OR FALSE?
If a matrix $A = (\begin{matrix} a & b \\ c & d \end{matrix})$ represents the orthogonal projection onto a line L , then the equation $a^{2} + b^{2} + c^{2} + d^{2} = 1$ must hold.

Short Answer

Expert verified

The given statement is false.

Step by step solution

01

Consider the parameters

The orthogonal projection of matrix is of form $(\begin{matrix} u_{1}^{2} & u_{1} u_{2} \\ u_{1} u_{2} & u_{2}^{2} \end{matrix})$ with $\overset{鈫�}{u} = [\begin{matrix} u_{1} \\ u_{2} \end{matrix}]$ . Thus, if

$a^{2} + b^{2} + c^{2} + d^{2} = 1 then$

${\begin{matrix} u_{1}^{4} + & (u_{1} u_{2}) \end{matrix}}^{2} + {(u_{1} u_{2})}^{2} + u_{2}^{4} = 1 鈬� {(u_{1} u_{2})}^{2} = 1$

Consider the matrix.

$A = (\begin{matrix} a & b \\ c & d \end{matrix})$

02

Find the values of the elements

If $A = (\begin{matrix} a & b \\ c & d \end{matrix})$ then,

The elements will be,

$a = u_{1}^{2}, d = u_{2}^{2}, b = c = u_{1} u_{2}$

Compute, $a^{2} + b^{2} + c^{2} + d^{2}$

$a^{2} + b^{2} + c^{2} + d^{2} = {(u_{1}^{2})}^{2} + {(u_{1} u_{2})}^{2} + {(u_{1} u_{2})}^{2} + {(u_{2}^{2})}^{2} 鈬� a^{2} + b^{2} + c^{2} + d^{2} = (u_{1}^{2} + u_{2}^{2}) (u_{1}^{2} + u_{2}^{2})$

Hence, the statement is false.

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Most popular questions from this chapter

Question 7: Suppose $\overset{鈫�}{v_{1}}, \overset{鈫�}{v_{2}}, \overset{鈫�}{v_{3}} 鈰� \overset{鈫�}{v_{m}}$ are arbitrary vectors in Consider the linear transformation from togiven by

$T [x_{1} x_{2} 鈰� x_{m}] = [x_{1} \overset{鈫�}{v_{1}}, x_{2} \overset{鈫�}{v_{2}}, x_{3} \overset{鈫�}{v_{3}} 鈰� x_{m} \overset{鈫�}{v_{m}}]$

If $A^{2}$ is invertible, then matrix A itself must be invertible.

TRUE OR FALSE?

There exists a real number k such that the matrix $[\begin{matrix} k - 1 & - 2 \\ - 4 & k - 3 \end{matrix}]$ fails to be invertible.

Consider an invertible $n 脳 n$ matrix Aand $a n n 脳 n$ matrix B.A certain sequence of elementary row operations transforms Ainto $I_{n}$ .

a. What do you get when you apply the same row operations in the same order to the matrix AB?

b. What do you get when you apply the same row operations to $I_{n}$ ?.

TRUE OR FALSE?

There exists an invertible $n 脳 n$ matrix with two identical rows.

See all solutions

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