Keyword Analysis & Research: how to show vectors are linearly independent

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Frequently Asked Questions

How to tell if vectors are dependent?

Vectors v1, . . . , vn are linearly dependent if the zero vector can be written as a nontrivial linear combination of the vectors: In this case, we refer to the linear combination as a linear dependency in v1, . . . , vn. On the other hand, if the only linear combination that equals the zero vector is the trivial linear combination, we say v1, . . . , vn are linearly independennonzero vectzero ...

Are two vectors having equal magnitude?

Two vectors are equal if they have the same length (magnitude) and direction. Examples: 1. Let u be the vector represented by the directed line segment from R = (-4, 2) to S = (-1, 6) a) Find the magnitude of u. b) Find the component form of the vector.

How do you find the linear combination of a vector?

We define a linear combination of vectors and examine whether a given vector may be expressed as a linear combination of other vectors, both algebraically and geometrically. A vector v is said to be a linear combination of vectors v 1, v 2, …, v n if v = a 1 v 1 + a 2 v 2 + … + a n v n for some scalars a 1, a 2, …, a n .

How to prove that two vectors are perpendicular?

Two vectors u= (a,b) and v= (c,d) in a coordinate plane are perpendicular if and only if their scalar product a*c + b*d is equal to zero: a*c + b*d = 0. Example 1 Prove that the vectors u= ( , ) and v= ( , ) are perpendicular. Solution The scalar product of these vectors is

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