Basis and Dimension of Vector Space Spanned by Vectors Example 2

VS07
Q Find dimension and basis of vector space spanned by following vectors
v1 = (1, 1, 0) ; v2 = (0, 1, 1) ; v3 = (2 3, 1) ; v4 = (1, 1, 1)

Theory

In this lecture we will learn about vector space and their properties.

Vector Space V – Span , Basis and Orthonormal basis
A collection of vectors which is closed under addition and scaler multiplication is called a vector space.

Closed under addition and scaler multiplication implies that linear combination of any number of vectors in the vector space also lies in the vector space.

Span – A collection of vectors v1, v2, …vk from vector space V is said to span V if every vector in V can be written as linear combination of v1, v2, …vk. (Note that v1, v2, …vk do not have to be linearly independent).

Basis – A collection of linearly independent vectors v1, v2, …vn that spans vector space V is called basis of vector space V. Basis for a vector space is not unique but number of vectors in the basis is fixed. This fixed number of vectors in basis is called dimension of vector space dim V or dim(V).

Every vector in vector space V can be written as linear combination of basis in a unique way. The scalers in the linear combination are called coordinates of the vector with respect to the basis.

Orthonormal basis – If the vectors in the basis are orthogonal to each other and have unit magnitude then the basis is called orthonormal basis. Any basis can be converted into orthonormal basis.

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Alpha Academy, Udaipur
Minakshi Porwal (9460189461)

[eunicedeepthivanka]

Sir what is the linear combination for V3 interns of v1, v2 and v4

[snehababuraj]

Why v3 column not considered as independent column nd basis too..

[sunainas]

Sir how to know the rank of 3×4 matrix... By indeterminate form... Please reply me