Q56E Find a basis 貜of 鈩�2聽such tha... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q56E (page 161)

Find a basis $貜$ of ${鈩�}^{2}$ such that localid="1660644788068" ${[\begin{matrix} 1 \\ 2 \end{matrix}]}_{貜} = [\begin{matrix} 3 \\ 5 \end{matrix}] and {[\begin{matrix} 3 \\ 4 \end{matrix}]}_{貜} = [\begin{matrix} 2 \\ 3 \end{matrix}]$ .

Short Answer

Expert verified

If $貜$ is ${鈩�}^{2}$ a basis such that $貜 = \{u, v\}$ and if

localid="1660646129256" ${[\begin{matrix} 1 \\ 2 \end{matrix}]}_{貜} = [\begin{matrix} 3 \\ 5 \end{matrix}] and {[\begin{matrix} 3 \\ 4 \end{matrix}]}_{貜} = [\begin{matrix} 2 \\ 3 \end{matrix}] . 鈬� \overset{鈫�}{u} = [\begin{matrix} 12 \\ 14 \end{matrix}] and \overset{鈫�}{v} = [\begin{matrix} - 7 \\ - 8 \end{matrix}]$

Step by step solution

Mentioning the concept

Consider a basis $貜 = \{{\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, {\overset{鈫�}{v}}_{3}, . . . ., {\overset{鈫�}{v}}_{n}\}$ of a subspace V of ${鈩�}^{n}$ Then any vector $X 鈭� {鈩�}^{n}$ can be written uniquely as-

$\overset{鈫�}{x} = c_{1} {\overset{鈫�}{v}}_{1} + c_{2} {\overset{鈫�}{v}}_{2} + . . . + c_{n} {\overset{鈫�}{v}}_{n}$

where the scalars $c_{1}, c_{2}, . . . ., c_{n}$ are called the $貜$ coordinates of vector $\overset{鈫�}{x}$ .

And the vector $[\begin{matrix} c_{1} \\ c_{2} \\ . \\ . \\ c_{n} \end{matrix}]$ is the $貜$ -coordinate vector of the vector $\overset{鈫�}{x}$ denoted by ${[\overset{鈫�}{x}]}_{貜}$

Also, $\overset{鈫�}{x} = S {[\overset{鈫�}{x}]}_{貜}$

Where $S = [{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2} {\overset{鈫�}{v}}_{3}, . . ., {\overset{鈫�}{v}}_{n}]$

Finding the vectors u→,v→

Since it is given that

${[\begin{matrix} 1 \\ 2 \end{matrix}]}_{貜} = [\begin{matrix} 3 \\ 5 \end{matrix}] and {[\begin{matrix} 3 \\ 4 \end{matrix}]}_{貜} = [\begin{matrix} 2 \\ 3 \end{matrix}] .$

Let,

$\overset{鈫�}{x} = [\begin{matrix} 1 \\ 2 \end{matrix}] and \overset{鈫�}{y} = [\begin{matrix} 3 \\ 4 \end{matrix}] 鈬� {[\overset{鈫�}{x}]}_{貜} = [\begin{matrix} 3 \\ 5 \end{matrix}] and {[\overset{鈫�}{y}]}_{貜} = [\begin{matrix} 2 \\ 3 \end{matrix}]$

And let the basis $貜 = \{\overset{鈫�}{u}, \overset{鈫�}{v}\}$ then we have

$[\begin{matrix} 1 \\ 2 \end{matrix}] = 3 [\begin{matrix} {\overset{鈫�}{u}}_{1} \\ {\overset{鈫�}{u}}_{2} \end{matrix}] + 5 [\begin{matrix} {\overset{鈫�}{v}}_{1} \\ {\overset{鈫�}{v}}_{2} \end{matrix}] and [\begin{matrix} 3 \\ 4 \end{matrix}] = 2 [\begin{matrix} {\overset{鈫�}{u}}_{1} \\ {\overset{鈫�}{u}}_{2} \end{matrix}] + 3 [\begin{matrix} {\overset{鈫�}{v}}_{1} \\ {\overset{鈫�}{v}}_{2} \end{matrix}] 鈬� 1 = 3 {\overset{鈫�}{u}}_{1} + 5 {\overset{鈫�}{v}}_{1} 2 = 3 {\overset{鈫�}{u}}_{2} + 5 {\overset{鈫�}{v}}_{2}$

and

$3 = 2 {\overset{鈫�}{u}}_{1} + 5 {\overset{鈫�}{v}}_{1} 4 = 2 {\overset{鈫�}{u}}_{2} + 5 {\overset{鈫�}{v}}_{2}$

$鈬� \overset{鈫�}{u} = [\begin{matrix} {\overset{鈫�}{u}}_{1} \\ {\overset{鈫�}{u}}_{2} \end{matrix}] = [\begin{matrix} 12 \\ 14 \end{matrix}] and \overset{鈫�}{v} = [\begin{matrix} {\overset{鈫�}{v}}_{1} \\ {\overset{鈫�}{v}}_{2} \end{matrix}] = [\begin{matrix} - 7 \\ - 8 \end{matrix}]$

Final Answer

If $貜$ of ${鈩�}^{2}$ is a basis such that $貜 = \{u, v\}$ and if

${[\begin{matrix} 1 \\ 2 \end{matrix}]}_{貜} = [\begin{matrix} 3 \\ 5 \end{matrix}] and {[\begin{matrix} 3 \\ 4 \end{matrix}]}_{貜} = [\begin{matrix} 2 \\ 3 \end{matrix}] . 鈬� \overset{鈫�}{u} = [\begin{matrix} 12 \\ 14 \end{matrix}] and \overset{鈫�}{v} = [\begin{matrix} - 7 \\ - 8 \end{matrix}]$

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91影视

Find a basis $貜$ of ${鈩�}^{2}$ such that localid="1660644788068" ${[\begin{matrix} 1 \\ 2 \end{matrix}]}_{貜} = [\begin{matrix} 3 \\ 5 \end{matrix}] and {[\begin{matrix} 3 \\ 4 \end{matrix}]}_{貜} = [\begin{matrix} 2 \\ 3 \end{matrix}]$ .

Short Answer

Step by step solution

Mentioning the concept

Finding the vectors u→,v→

Final Answer

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91影视

Find a basis 貜of 鈩�2such that localid="1660644788068" [12]貜=[35]and[34]貜=[23].

Short Answer

Step by step solution

Mentioning the concept

Finding the vectors u→,v→

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Probability and Statistics

Mechanics Maths

Discrete Mathematics

Logic and Functions

Applied Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Find a basis $貜$ of ${鈩�}^{2}$ such that localid="1660644788068" ${[\begin{matrix} 1 \\ 2 \end{matrix}]}_{貜} = [\begin{matrix} 3 \\ 5 \end{matrix}] and {[\begin{matrix} 3 \\ 4 \end{matrix}]}_{貜} = [\begin{matrix} 2 \\ 3 \end{matrix}]$ .