# Linear algebra basis/dimensions

• rosh300
In summary, for part a) the dimension of vector space V is N and the basis is all polynomials over R of degree at most n and whose coefficients add to 0. For part b) the dimension of vector space V is 2 and the basis is functions df/dx-3 and df/dx+3.
rosh300

## Homework Statement

Find the dimensions and basis of the following vector space V over the given field K:
a) V is the set of all polynomials over R (real) of degree at most n and whose coefficients add to 0, K = R (real numbers)
b) K = R (real), and V is the set of functions from R to R which are solutions of the differential equation: d2f/dx2 - 9f = 0

## Homework Equations

definition of basis, spanning and dimensions

## The Attempt at a Solution

for part a) i think the dimetions is N and the basis is all the factors

for part b) i think it has 2 dimentions and the basis is df/dx - 3 and df/dx +3

Sort of. But what do you mean by 'factors' and df/dx-3 isn't a function from R->R. It's a differential equation. Can you spell out exactly what a basis is in each case?

I might be wrong, but it appears that you "factored" this differential equation:
d2f/dx2 - 9f = 0

to get this:
df/dx - 3 and df/dx +3
One problem with that is that the first DE had -9f, not -9.

Dick said:
Sort of. But what do you mean by 'factors' and df/dx-3 isn't a function from R->R. It's a differential equation. Can you spell out exactly what a basis is in each case?

By basic i mean something which is linear independent and spans V

and by factors i mean suppose a1, a2, a3 ... an were the roots, then the factors would be (x - a1), (x - a2), (x - a3) ... (x - an) I suppose i should specfy the Real roots

rosh300 said:
By basic i mean something which is linear independent and spans V

and by factors i mean suppose a1, a2, a3 ... an were the roots, then the factors would be (x - a1), (x - a2), (x - a3) ... (x - an) I suppose i should specfy the Real roots

Exactly. In the first case you should specify N polynomials and in the second case you should specify two functions.

## 1. What is a basis in linear algebra?

A basis in linear algebra refers to a set of linearly independent vectors that span a vector space. This means that any vector in the vector space can be expressed as a linear combination of the basis vectors. The number of basis vectors in a basis is called the dimension of the vector space.

## 2. How do you determine the dimension of a vector space?

The dimension of a vector space is equal to the number of vectors in a basis for that vector space. To determine the dimension, you can find a set of linearly independent vectors that span the vector space and count the number of vectors in that set.

## 3. Can a vector space have more than one basis?

Yes, a vector space can have multiple bases. However, all bases for a given vector space will have the same number of vectors, which is equal to the dimension of the vector space. Additionally, any vector in the vector space can be expressed as a linear combination of the vectors in any basis for that vector space.

## 4. How do you find the basis for a vector space?

To find a basis for a vector space, you can start by finding a set of linearly independent vectors that span the vector space. This can be done by solving a system of linear equations or using other methods such as Gaussian elimination. Once you have a set of linearly independent vectors, you can check if they span the entire vector space. If they do, then you have found a basis for the vector space.

## 5. What is the relationship between basis and dimension in linear algebra?

The basis and dimension of a vector space are closely related. The basis of a vector space is a set of linearly independent vectors that span the vector space, and the dimension is the number of vectors in that basis. The dimension of a vector space is also the maximum number of linearly independent vectors that can exist in that vector space. Therefore, the basis and dimension provide important information about the structure and size of a vector space.

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