Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 3: Q19E (page 164)

If vectors ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, . . . {\overset{鈫�}{v}}_{n}$ span $R^{4},$ then 鈥榥鈥� must be equal to 4.

Short Answer

Expert verified

The above statement is false.

If vectors ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, . . . {\overset{鈫�}{v}}_{n}$ span $R^{4},$ then 鈥榥鈥� may or may not be equal to 4.

Step by step solution

Finding the vectors v→1,v→2,...v→n that span R4

Let us assume the set of vectors

$S = \{{\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, {\overset{鈫�}{v}}_{3}, {\overset{鈫�}{v}}_{4}, {\overset{鈫�}{v}}_{5}\}$

Such that,

${\overset{鈫�}{v}}_{1} = [\begin{matrix} 0 \\ 0 \\ 1 \\ 1 \end{matrix}], {\overset{鈫�}{v}}_{2} = [\begin{matrix} - 1 \\ 1 \\ 1 \\ 2 \end{matrix}], {\overset{鈫�}{v}}_{3} = [\begin{matrix} 1 \\ 1 \\ 0 \\ 0 \end{matrix}], {\overset{鈫�}{v}}_{4} = [\begin{matrix} 2 \\ 1 \\ 2 \\ 1 \end{matrix}], {\overset{鈫�}{v}}_{5} = [\begin{matrix} 1 \\ - 1 \\ 0 \\ - 1 \end{matrix}]$

To prove v→1,v→2,v→3,v→4,v→5 spans R4

Let us write the above column vectors in the matrix

$A = [\begin{matrix} 0 & - 1 & 1 & 2 & 1 \\ 0 & 1 & 1 & 1 & - 1 \\ 1 & 1 & 0 & 2 & 0 \\ 1 & 2 & 0 & 1 & - 1 \end{matrix}]$

$鈬� A = [\begin{matrix} 1 & 2 & 0 & 1 & - 1 \\ 0 & 1 & 1 & 1 & - 1 \\ 1 & 1 & 0 & 2 & 0 \\ 0 & - 1 & 1 & 2 & 1 \end{matrix}] (R_{4} 鈫� R_{1})$

$鈬� A = [\begin{matrix} 1 & 2 & 0 & 1 & - 1 \\ 0 & 1 & 1 & 1 & - 1 \\ 0 & - 1 & 0 & 1 & 1 \\ 0 & - 1 & 1 & 2 & 1 \end{matrix}] (R_{3} 鈫� R_{3} - R_{1})$

$鈬� A = [\begin{matrix} 1 & 2 & 0 & 1 & - 1 \\ 0 & 1 & 1 & 1 & - 1 \\ 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 2 & 3 & 0 \end{matrix}] (R_{3} 鈫� R_{3} + R_{2}, R_{4} 鈫� R_{4} + R_{2})$

$鈬� A = [\begin{matrix} 1 & 2 & 0 & 1 & - 1 \\ 0 & 1 & 1 & 1 & - 1 \\ 0 & 0 & 1 & 2 & 0 \\ 0 & 0 & 0 & - 1 & 0 \end{matrix}] (R_{4} 鈫� R_{4} - 2 R_{3})$

To show the set S=v→1,v→2,v→3,v→4,v→5 spans R4

Since, there are four pivot elements,

Therefore,

$s p a n S = R^{4}$

Thus, we see that vectors ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, {\overset{鈫�}{v}}_{3}, {\overset{鈫�}{v}}_{4} and {\overset{鈫�}{v}}_{5}$ spans $R^{4},$ but n= 5.

Final Answer

We have the set of vectors $S = \{{\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, {\overset{鈫�}{v}}_{3}, {\overset{鈫�}{v}}_{4}, {\overset{鈫�}{v}}_{5}\}$ that spans $R^{4},$ has number of vectors 5.

Hence, the statement, 鈥淚f vectors ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, . . . {\overset{鈫�}{v}}_{n}$ span $R^{4},$ then 鈥榥鈥� must be equal to 4鈥� is false.

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If vectors ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, . . . {\overset{鈫�}{v}}_{n}$ span $R^{4},$ then 鈥榥鈥� must be equal to 4.

Short Answer

Step by step solution

Finding the vectors v→1,v→2,...v→n that span R4

To prove v→1,v→2,v→3,v→4,v→5 spans R4

To show the set S=v→1,v→2,v→3,v→4,v→5 spans R4

Final Answer

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91影视

If vectors v鈫�1,v鈫�2,...v鈫�nspan R4,then 鈥榥鈥� must be equal to 4.

Short Answer

Step by step solution

Finding the vectors v→1,v→2,...v→n that span R4

To prove v→1,v→2,v→3,v→4,v→5 spans R4

To show the set S=v→1,v→2,v→3,v→4,v→5 spans R4

Final Answer

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Geometry

Logic and Functions

Mechanics Maths

Discrete Mathematics

Probability and Statistics

Study anywhere. Anytime. Across all devices.

If vectors ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2}, . . . {\overset{鈫�}{v}}_{n}$ span $R^{4},$ then 鈥榥鈥� must be equal to 4.