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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 4: Q54E (page 197)

In Exercises 54 through 58, let V be the plane with equation. $x_{1} + 2 x_{2} + 3 x_{3} = 0 i n R^{3}$ In each exercise, find the matrix B of the given transformation T from V to V, with respect to the basis $[\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix}] [\begin{matrix} 5 \\ - 4 \\ 1 \end{matrix}]$ Note that the domain and target space of T are restricted to the plane V, so that B will be a $2 脳 2$ matrix.
54. The orthogonal projection onto the line spanned by vector $[\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix}]$

Short Answer

Expert verified

The solution is $B = (\begin{matrix} 1 & - 4 \\ 1 & 5 \end{matrix})$

Step by step solution

01

Step 1:Solution for the matrix of the given transformation T

Consider the matrix of the given transformation be T

Let V be the plane with equation $x_{1} + 2 x_{2} - 3 x_{3} = 0$ in R³.

Also we know that the domain and the target space of T are restricted to the plane V so that B will be a $2 脳 2$ matrix.

Assume $f (x) = a c o s (x) + b s i n (x)$ be the function

Hence the basis of the equation be $[\begin{matrix} 1 \\ 1 \\ - 1 \end{matrix}] [\begin{matrix} 5 \\ - 4 \\ 1 \end{matrix}]$

Then, the two equation be as follows.

${\overset{鈫�}{b}}_{1} = a_{1} + a_{2} \overset{鈫�}{b} {}_{2}= - 4 a_{1} + 5 a_{2}$

Therefore the matrix be B as follows

$B = (\begin{matrix} 1 & - 4 \\ 1 & 5 \end{matrix})$

Hence the solution.

02

Step 2:Solution for the isomorphism of the given transformation T

Consider $B = (\begin{matrix} 1 & - 4 \\ 1 & 5 \end{matrix})$ be the matrix of the given transformation T.

$[B] = d e t (B) 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� = a d - b c 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� = 5 + 4 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� 鈥� = 9 鈮� 0$

Thus the transformation T is invertible as well.So T is an isomorphism.

Hence the solution.

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Most popular questions from this chapter

Show that if 0 is the neutral element of a linear space V then k0=0, for all scalars k.

T denotes the space of infinity sequence of real numbers, $T (f (t)) = [\begin{matrix} f (5) \\ f (7) \\ f (11) \end{matrix}] f r o m P_{2} t o R^{3} .$ .

Find the basis of all real linear space $C^{2}$ , and determine its dimension.

Find the transformation is linear and determine whether the transformation is an isomorphism.

Which of the subsets Vof ${鈩�}^{3 脳 3}$ given in Exercise 6through 11are subspaces of ${鈩�}^{3 脳 3}$ . The $3 脳 3$ matrices Asuch that vector $[\begin{matrix} 1 \\ 2 \\ 3 \end{matrix}]$ is in the kernel of A.

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