Dimension and basis for solution space of a homogeneous system of equations Example 1

VS11
Find the dimension and basis of vectors space spanned by solutions of homogeneous system of equations
x+2y = 0
y+z = 0

in other words
Find the dimension and bais for the vector space V={(x, y, z) in R3 : x+2y = 0 , y+z = 0 }

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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Minakshi Porwal (9460189461)

[BushraImran-zn]

The value you assign like alpha minus alpha is random??

[eunicedeepthivanka]

Sir what do you mean by x, y, z belonging to R^3