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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: Q48E (page 160)

In the accompanying figure, sketch the vector $\overset{鈫�}{x}$ with ${[\overset{鈫�}{x}]}_{貜} = [\begin{matrix} - 1 \\ 2 \end{matrix}]$ , where $貜$ is the basis of ${鈩�}^{2}$ consisting of the vectors $\overset{鈫�}{v}, \overset{鈫�}{w}$ .

Short Answer

Expert verified

If ${[\overset{鈫�}{x}]}_{貜} = [\begin{matrix} - 1 \\ 2 \end{matrix}]$ where $貜$ is the basis of ${鈩�}^{2}$ consisting of the vectors .

Then the graph of $\overset{鈫�}{x} = - 1 \overset{鈫�}{v} + 2 \overset{鈫�}{w}$ is

Step by step solution

01

Sketching the vectors v→,w→

02

Sketching the vectors -1v→,2w→

Since we have given that ${[\overset{鈫�}{x}]}_{貜} = [\begin{matrix} - 1 \\ 2 \end{matrix}]$ , then the vector $\overset{鈫�}{x} = - 1 \overset{鈫�}{v} + 2 \overset{鈫�}{w}$

03

Sketching the vector x→=-1v→+2w→

04

Final Answer

If ${[\overset{鈫�}{x}]}_{貜} = [\begin{matrix} - 1 \\ 2 \end{matrix}]$ where $貜$ is the basis of ${鈩�}^{2}$ consisting of the vectorslocalid="1660377766283" $\overset{鈫�}{v}, \overset{鈫�}{w}$ .

Then the graph oflocalid="1660377771570" $\overset{鈫�}{x} = - 1 \overset{鈫�}{y} + 2 \overset{鈫�}{w}$ is

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Most popular questions from this chapter

Consider the matrices

$A = [\begin{matrix} 1 & 0 & 2 & 0 & 4 & 0 \\ 0 & 1 & 3 & 0 & 5 & 0 \\ 0 & 0 & 0 & 1 & 6 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] and B = [\begin{matrix} 1 & 0 & 2 & 0 & 4 & 0 \\ 0 & 1 & 3 & 0 & 5 & 0 \\ 0 & 0 & 0 & 1 & 7 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}]$

Show that the kernels of the matrices A and B are different

Show that if a 3 x 3 matrix A represents the reflection about a plane, then A is similar to the matrix $[\begin{matrix} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & - 1 \end{matrix}]$ .

Consider a nonzero vector $\overset{鈫�}{蠀}$ in ${鈩�}^{3}$ . Using a geometric argument, describe the kernel of the linear transformation $T$ from ${鈩�}^{3}$ to ${鈩�}^{3}$ given by,

$T (\overset{鈫�}{x}) = \overset{鈫�}{蠀} 脳 \overset{鈫�}{x}$

See Definition A.9 in the Appendix.

In Exercises 1 through 20, find the redundant column vectors of the given matrix A 鈥渂y inspection.鈥� Then find a basis of the image of A and a basis of the kernel of A.

20. $[\begin{matrix} 1 & 0 & 5 & 3 & - 3 \\ 0 & 0 & 0 & 1 & 3 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \end{matrix}]$

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.

53. $(0, 0), (1, 0), (2, 0), (0, 1), (1, 1), (2, 1), (0, 2), (1, 2), (3, 2)$ .

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