Q53E Consider the basisI聽of聽鈩�2聽c... [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: Q53E (page 161)

Consider the basis $I of {鈩�}^{2}$ consisting of the vectors $[\begin{matrix} 1 \\ 2 \end{matrix}] and [\begin{matrix} 3 \\ 4 \end{matrix}]$
.We are told that ${[\overset{鈬赌}{x}]}_{I} = [\begin{matrix} 7 \\ 11 \end{matrix}]$ for a certain vector $\overset{鈬赌}{x} in {鈩�}^{2}$
Find

Short Answer

Expert verified

If, $I o f {鈩�}^{2}$ is a basis consisting of the vectors $[\begin{matrix} 1 \\ 2 \end{matrix}] and [\begin{matrix} 3 \\ 4 \end{matrix}] and {[\overset{鈬赌}{x}]}_{I} = [\begin{matrix} 7 \\ 11 \end{matrix}]$ for a certain vector $\overset{鈬赌}{x} in {鈩�}^{2}$

Then, $\overset{鈬赌}{x} = [\begin{matrix} 40 \\ 58 \end{matrix}]$

Step by step solution

Mentioning the concept

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

role="math" localid="1660714253890" $\overset{鈬赌}{x} = c_{1} {\overset{鈬赌}{v}}_{1} + c_{2} {\overset{鈬赌}{v}}_{2} + . . . c_{n} {\overset{鈬赌}{v}}_{n} where the scalars c_{1}, c_{2}, . . ., c_{n} are called the I coordinates of vector \overset{鈬赌}{x}$

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

$Also, \overset{鈬赌}{x} = S {[\overset{鈬赌}{x}]}_{I} Where S = [{\overset{鈬赌}{v}}_{1} {\overset{鈬赌}{v}}_{2} {\overset{鈬赌}{v}}_{3}, . . . ., {\overset{鈬赌}{v}}_{n}]$

Finding the vector x⇀

$We have given a basis I = \{{\overset{鈬赌}{v}}_{1}, {\overset{鈬赌}{v}}_{2}\} where {\overset{鈬赌}{v}}_{1} = [\begin{matrix} 1 \\ 2 \end{matrix}] and {\overset{鈬赌}{v}}_{2} = [\begin{matrix} 3 \\ 4 \end{matrix}] . Then \overset{鈬赌}{x} = S {[\overset{鈬赌}{x}]}_{I} 鈭� {[\overset{鈬赌}{x}]}_{I} = [\begin{matrix} 7 \\ 11 \end{matrix}] and S = [{\overset{鈬赌}{v}}_{1} {\overset{鈬赌}{v}}_{2}] 鈬� S = [\begin{matrix} 1 & 3 \\ 2 & 4 \end{matrix}] 鈬� \overset{鈬赌}{x} = [\begin{matrix} 1 & 3 \\ 2 & 4 \end{matrix}] [\begin{matrix} 7 \\ 11 \end{matrix}] 鈬� \overset{鈬赌}{x} = [\begin{matrix} 7 + 33 \\ 14 + 44 \end{matrix}] 鈬� \overset{鈬赌}{x} = [\begin{matrix} 40 \\ 58 \end{matrix}]$

Final Answer

$If, I o f {鈩�}^{2} is a basis consisting of the vectors [\begin{matrix} 1 \\ 2 \end{matrix}] and [\begin{matrix} 3 \\ 4 \end{matrix}] and {[\overset{鈬赌}{x}]}_{I} = [\begin{matrix} 7 \\ 11 \end{matrix}] for a certain vector \overset{鈬赌}{x} in {鈩�}^{2} Then, \overset{鈬赌}{x} = [\begin{matrix} 40 \\ 58 \end{matrix}]$

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91影视

Consider the basis $I of {鈩�}^{2}$ consisting of the vectors $[\begin{matrix} 1 \\ 2 \end{matrix}] and [\begin{matrix} 3 \\ 4 \end{matrix}]$
.We are told that ${[\overset{鈬赌}{x}]}_{I} = [\begin{matrix} 7 \\ 11 \end{matrix}]$ for a certain vector $\overset{鈬赌}{x} in {鈩�}^{2}$
Find

Short Answer

Step by step solution

Mentioning the concept

Finding the vector x⇀

Final Answer

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91影视

Consider the basisIof鈩�2consisting of the vectors[12]and[34].We are told that[x鈬赌]I=[711]for a certain vectorx鈬赌in鈩�2Find

Short Answer

Step by step solution

Mentioning the concept

Finding the vector x⇀

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Calculus

Probability and Statistics

Geometry

Discrete Mathematics

Decision Maths

Study anywhere. Anytime. Across all devices.

Consider the basis $I of {鈩�}^{2}$ consisting of the vectors $[\begin{matrix} 1 \\ 2 \end{matrix}] and [\begin{matrix} 3 \\ 4 \end{matrix}]$
.We are told that ${[\overset{鈬赌}{x}]}_{I} = [\begin{matrix} 7 \\ 11 \end{matrix}]$ for a certain vector $\overset{鈬赌}{x} in {鈩�}^{2}$
Find