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For each set, determine if it forms a vector space. If it does, find a basis, state its dimension, and find the vector corresponding to ([1, 0]).

User Whi
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Final answer:

A set forms a vector space if it satisfies all vector space axioms. To find a basis for a two-dimensional vector space, the horizontal and vertical components of the vectors are determined, and the vectors are expressed in terms of the unit vectors i and j. The dimension is the number of vectors in a basis.

Step-by-step explanation:

To determine if a set of vectors forms a vector space, one must verify if it satisfies all vector space axioms such as closure under addition and scalar multiplication, having a zero vector, and others. If a set forms a vector space, we can find a basis for it, which is a set of linearly independent vectors that span the entire space. The dimension of the vector space is the number of vectors in a basis. To find the vector corresponding to ([1, 0]), we need to align it with the defined coordinate system.

First, we'll determine the coordinate system for the vectors in question. For a two-dimensional vector problem, a convenient coordinate system is one with a horizontal x-axis and a vertical y-axis. The horizontal and vertical components of a vector A in this coordinate system can be described using equations such as Ax = A cos θ and Ay = A sin θ, where θ is the angle between vector A and the horizontal axis.

When we have two vectors that are not parallel, we can simply project these onto the x and y axes to obtain their components. We can then express these vectors in terms of the unit vectors i and j, which are aligned with the x-axis and y-axis respectively. For higher dimensions, vectors would respectively have more components.

Analyzing one-dimensional and two-dimensional relative motion problems using the position and velocity vector equations also involves working with vectors in a coordinate system and solving for their components and magnitudes.

User Harpax
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