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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: Q38E (page 160)

In Exercises 37 through 42 , find a basis $I o f {鈩�}^{n}$ such that the $I - m a t r i x$ B of the given linear transformation T is diagonal.
Orthogonal projection T onto the line in ${鈩�}^{n}$ spanned by $[\begin{matrix} 2 \\ 3 \end{matrix}]$ .

Short Answer

Expert verified

The matrix is, $B = [\begin{matrix} 2 \\ 3 \end{matrix}] [\begin{matrix} 3 \\ - 2 \end{matrix}]$

Step by step solution

01

Consider the vector.

The vector is,

${\overset{鈫�}{v}}_{1} = [\begin{matrix} 2 \\ 3 \end{matrix}]$

Consider the other vector ${\overset{鈫�}{v}}_{2}$ that is perpendicular to ${\overset{鈫�}{v}}_{1}$ to obtain the other matrix of the form:

$B = [\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix}]$

02

Compute the matrix using linear combination.

Consider the vector

${\overset{鈫�}{v}}_{1} = [\begin{matrix} 2 \\ 3 \end{matrix}]$

The orthogonal vector will be,

localid="1660636146637" ${\overset{鈫�}{v}}_{2} = [\begin{matrix} 3 \\ - 2 \end{matrix}]$

The matrix will be,

localid="1660636098121" $鈭� B = [\begin{matrix} 2 \\ 3 \end{matrix}] [\begin{matrix} 3 \\ - 2 \end{matrix}]$

03

Final answer.

The matrix is, $B = [\begin{matrix} 2 \\ 3 \end{matrix}] [\begin{matrix} 3 \\ - 2 \end{matrix}]$ .

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Most popular questions from this chapter

Consider a subspace $V$ in ${鈩�}^{m}$ that is defined by homogeneous linear equations

$| \begin{array}{l} a_{11} x_{1} + a_{12} x_{2} + 鈥� + a_{1 m} x_{m} = 0 \\ a_{21} x_{1} + a_{22} x_{2} + 鈥� + a_{2 m} x_{m} = 0 \\ 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� 鈰� 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� 鈰� 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆€夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆€� 鈰� 鈥夆赌夆赌夆赌夆赌夆赌夆赌夆赌夆赌� 鈰� 鈥� \\ a_{n 1} x_{1} + a_{22} x_{2} + 鈥� + a_{nm} x_{m} = 0 \end{array} |$ .

What is the relationship between the dimension of $V$ and the quantity $m - n$

? State your answer as an inequality. Explain carefully.

In Problem 46 through 55, Find all the cubics through the given points. You may use the results from Exercises 44 and 45 throughout. If there is a unique cubic, make a rough sketch of it. If there are infinitely many cubics, sketch two of them.

54. $(0, 0) (1, 0) (2, 0) (0, 1) (1, 1) (2, 1) (0, 2) (1, 2) (2, 2)$ .

In Exercises 25through 30, find the matrix Bof the linear transformation $T (\overset{鈫�}{x}) = A (\overset{鈫�}{x})$ with respect to the basis $J = ({\overset{鈫�}{v}}_{1}, 鈥� . ., {\overset{鈫�}{v}}_{m})$ .

$A = (\begin{matrix} 0 & 1 \\ 2 & 3 \end{matrix}); {\overset{鈫�}{v}}_{1} = [\begin{matrix} 1 \\ 2 \end{matrix}], {\overset{鈫�}{v}}_{2} = [\begin{matrix} 1 \\ 1 \end{matrix}]$

Consider the plane $x_{1} + 2 x_{2} + x_{3} = 0$ with basis B consisting of vectors $[\begin{matrix} - 1 \\ 0 \\ 1 \end{matrix}] and [\begin{matrix} - 2 \\ 1 \\ 0 \end{matrix}]$ . If ${[\overset{鈫�}{x}]}_{B} = [\begin{matrix} 2 \\ - 3 \end{matrix}]$

Consider a non-zero vector $\overset{鈫�}{蠀}$ in $\overset{}{{鈩�}^{n}}$ . What is the dimension of the space of all vectors in ${鈩�}^{n}$ that are perpendicular to $\overset{鈫�}{蠀}$ ?

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