What is the basis for the given subspace in R^5?

In summary, a subspace is a subset of a vector space that obeys the properties of a vector space. Its basis is a set of linearly independent vectors that span the subspace, allowing for calculations and transformations. The basis can be found through methods such as Gaussian elimination or the Gram-Schmidt process, and its dimension is equal to the number of vectors in the basis.
  • #1
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Homework Statement



find a basis for the subspace R^5 that consists of all the vectors of the form [(b-c), (d-2b), (4d), (c-2d), (6d+2b)]

Homework Equations





The Attempt at a Solution



the only solution I can think of is e1, e2, e3, e4, e5... I don't think it's that simple though... would appreciate any input on this question. Thanks!
 
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  • #2
Code:
a = b - c
b = -2b   + d
c =         4d
d =      c - 2d
e = 2b     + 6d
A vector in this subspace is a linear combination of three vectors, which you can pick out of the equations above.
 

What is a subspace?

A subspace is a subset of a vector space that satisfies the properties of a vector space. In other words, it is a subset of a larger space that is closed under vector addition and scalar multiplication.

What is the basis of a subspace?

The basis of a subspace is a set of linearly independent vectors that span the subspace. In other words, the basis is the minimum number of vectors needed to represent all other vectors in the subspace through linear combinations.

Why is finding a basis for a subspace important?

Finding a basis for a subspace is important because it allows us to understand the structure of the subspace and to perform calculations and transformations on the subspace. It also allows us to find a minimal set of vectors needed to represent the subspace, which can be useful in various applications.

How do you find the basis for a subspace?

To find the basis for a subspace, we need to find a set of linearly independent vectors that span the subspace. This can be done through various methods, such as Gaussian elimination, finding the null space of a matrix, or using the Gram-Schmidt process.

What is the relationship between the dimension of a subspace and its basis?

The dimension of a subspace is equal to the number of vectors in its basis. This means that the basis of a subspace will always have the same number of vectors as the dimension of the subspace. Additionally, any set of linearly independent vectors that span the subspace can be considered a basis for that subspace.

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