Q16E Vectors 100,聽213,聽321聽 form a... [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: Q16E (page 164)

Vectors $[\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ 1 \\ 3 \end{matrix}], [\begin{matrix} 3 \\ 2 \\ 1 \end{matrix}]$ form a basis of $R^{3}$ .

Short Answer

Expert verified

The above statement is true.

Vectors $[\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ 1 \\ 3 \end{matrix}], [\begin{matrix} 3 \\ 2 \\ 1 \end{matrix}]$ form a basis of $R^{3}$ .

Step by step solution

Definition of a basis

Any subset S of a vector space V(K) is called a basis of V(K), if -

S is linearly independent.
S generates V

i.e. L(S) = V

To check whether the given vectors form a basis of R3

Let $S = \{[\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ 1 \\ 3 \end{matrix}], [\begin{matrix} 3 \\ 2 \\ 1 \end{matrix}]\}$

Again let, $a_{1}, a_{2}, a_{3} 鈭� R^{3}$ be such that

$a_{1} [\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}] + a_{2} [\begin{matrix} 2 \\ 1 \\ 3 \end{matrix}] + a_{3} [\begin{matrix} 3 \\ 2 \\ 1 \end{matrix}] = [\begin{matrix} 0 \\ 0 \\ 0 \end{matrix}]$

$鈬� a_{1} + 2 a_{2} + 3 a_{3} = 0 a_{2} + 2 a_{3} = 0 a_{3} = 0$

$鈬� a_{1} = 0, a_{2} = 0, a_{3} = 0$

Hence, the vectors are L.I. and the only solution to these equations is

$a_{1} = 0, a_{2} = 0, a_{3} = 0$

Final Answer

For a set $S = \{[\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ 1 \\ 3 \end{matrix}], [\begin{matrix} 3 \\ 2 \\ 1 \end{matrix}]\},$ we can clearly see that the vectors of S are L.I. and

$L (S) = R^{3} .$

Therefore,

The vectors $[\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ 1 \\ 3 \end{matrix}], [\begin{matrix} 3 \\ 2 \\ 1 \end{matrix}]$ form a basis of $R^{3}$ .

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91影视

Vectors $[\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ 1 \\ 3 \end{matrix}], [\begin{matrix} 3 \\ 2 \\ 1 \end{matrix}]$ form a basis of $R^{3}$ .

Short Answer

Step by step solution

Definition of a basis

To check whether the given vectors form a basis of R3

Final Answer

One App. One Place for Learning.

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91影视

Vectors 100,213,321 form a basis ofR3 .

Short Answer

Step by step solution

Definition of a basis

To check whether the given vectors form a basis of R3

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Theoretical and Mathematical Physics

Discrete Mathematics

Applied Mathematics

Geometry

Pure Maths

Study anywhere. Anytime. Across all devices.

Vectors $[\begin{matrix} 1 \\ 0 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ 1 \\ 3 \end{matrix}], [\begin{matrix} 3 \\ 2 \\ 1 \end{matrix}]$ form a basis of $R^{3}$ .