# Finding a Basis for R4 with Given Data

• Dell
In summary, when finding a basis for a given subspace, you can start by using the given conditions to find a set of linearly independent vectors. Then, to complete the basis for the entire space, you can use the Gram-Schmidt process to find additional vectors that are independent from the original set. It is not meaningful to compare two separate bases for the same subspace in terms of linear independence, as a basis must be the largest set of linearly independent vectors that span the subspace.
Dell
i am given a subspace of R4 W={(a b c d)} and know a+c=0 c-2d=0, and am asked to find a basis for W,
i wrote (-2 0 2 1)(0 1 0 0),
now i am asked to find the missing vectors so that the new basis will be a basis for R4. to find this i need vectors that are independant and are the basis for R4-W? how can i do this??
just a guess would be (-2 0 2 1)(0 1 0 0)(0 0 1 0)(0 0 0 1) but that's just because i know that those four would cover the whole space and are independant. what is the mathematical way of solving this?

Gram-Schmidt process --http://en.wikipedia.org/wiki/Gram_schmidt

another question, not related,
is it possible for 2 separate basises of a subspace to be linearly independant of one another, or do they always need to be dependant

I don't think it's meaningful to talk about one basis being independent of another. For linear independence/dependence, we're always talking about a collection of vectors.

Suppose v1, v2, ... , vn are a basis for a subspace W and u1, u2, ..., un are another basis for W. The set of vectors {v1, v2, ..., vn, u1} has to be a linearly dependent set, meaning that u1 has to be a linear combination of v1, v2, ... , vn.

A basis for a subspace W is the largest set of vectors that a) is linearly independent, and b) spans W. If you add any vector to this basis, the added vector must be a linear combination of the original basis vectors.

## What is a basis for R4?

A basis for R4 is a set of four vectors that span the entire four-dimensional space of R4.

## How do you find a basis for R4 with given data?

To find a basis for R4 with given data, you can use the Gaussian elimination method to reduce the given vectors to their row echelon form and then choose the linearly independent vectors as the basis.

## What does it mean for vectors to be linearly independent?

Vectors are linearly independent if none of them can be expressed as a linear combination of the others. In other words, no vector in the set can be written as a linear combination of the remaining vectors.

## How many vectors are needed to form a basis for R4?

Since R4 is a four-dimensional space, a basis for R4 must contain four linearly independent vectors.

## Why is finding a basis for R4 important?

Finding a basis for R4 is important because it allows us to easily represent and manipulate four-dimensional data in a more manageable way. It also helps in solving systems of linear equations and performing transformations in four-dimensional space.

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