Q9E If 聽v鈫�1 and聽 v鈫�2聽are line... [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 6: Q9E (page 306)

If ${\overset{鈫�}{v}}_{1}$ and ${\overset{鈫�}{v}}_{2}$ are linearly independent vectors in $R^{2}$ , what is the relationship between $[{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2}]$ and $[{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2}^{鈯�}]$ , where ${\overset{鈫�}{v}}_{2}^{鈯�}$ is the component of ${\overset{鈫�}{v}}_{2}$ orthogonal to ${\overset{鈫�}{v}}_{1}$ ?

Short Answer

Expert verified

Therefore, the relationship between $[{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2}]$ and $[{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2}^{鈯�}]$ is given by,

$\det [{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2}^{鈯�}] = \det [{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2}] 肠辞蝉蠁$

Step by step solution

Definition.

A determinant is a unique number associatedwith a square matrix.

A determinant is a scalar value that is a function of the entries of a square matrix.

It is the signed factor by which areas are scaled by this matrix. If the sign is negative the matrix reverses orientation.

To find the relationship between det[v→1 v→2] and det[v→1 v→2⊥] .

If ${\overset{鈫�}{v}}_{1}, {\overset{鈫�}{v}}_{2} 鈭� {鈩�}^{2}$ are linearly independent vectors, then $[{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2}] 鈮� 0$ .

If $蠁$ is the angle between ${\overset{鈫�}{v}}_{1}$ and ${\overset{鈫�}{v}}_{2}$ , then the angle between ${\overset{鈫�}{v}}_{1}$ and ${\overset{鈫�}{v}}_{2} 鈯�$ is $\frac{蟺}{2} - 蠁$ . Therefore,

$\det [{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2}^{鈯�}] = \det [{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2}] 肠辞蝉蠁$

Therefore, the relationship between $\det [{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2}]$ and $\det [{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2}^{鈯�}]$ is .

$\det [{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2}^{鈯�}] = \det [{\overset{鈫�}{v}}_{1} {\overset{鈫�}{v}}_{2}] 肠辞蝉蠁$

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Short Answer

Step by step solution

Definition.

To find the relationship between det[v→1 v→2] and det[v→1 v→2⊥] .

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91影视

If v鈫�1 and v鈫�2are linearly independent vectors inR2 , what is the relationship between [v鈫�1v鈫�2] and [v鈫�1v鈫�2鈯�], where v鈫�2鈯� is the component of v鈫�2 orthogonal tov鈫�1 ?

Short Answer

Step by step solution

Definition.

To find the relationship between det[v→1 v→2] and det[v→1 v→2⊥] .

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Statistics

Decision Maths

Geometry

Applied Mathematics

Study anywhere. Anytime. Across all devices.