Sage-Tips

Is the subset of polynomials a subspace of the vector space of all polynomials?

The subset Pn of polynomials of degree at most n is a subspace of P. Well, H contains 0, scalar multiples of elements of H are in H, and the sum of any two vectors in H is in H, and so H is a subspace of R3.

Can any vector be expressed as a linear combination?

Yes . Remember that, in general, a vector may be treated as an element of a Vector – space and therefore by definition, every vector of it can always be expressed as a linear combination of its basis vectors . Take a simple example of the vector- space R^(3) over R. A basis of R^(3) is S = {(1,0,0), (0,1,0), (0,0,1)}.

What makes something a linear combination?

In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).

What is a linear combination of a matrix?

A matrix is a linear combination of if and only if there exist scalars , called coefficients of the linear combination, such that. In other words, if you take a set of matrices, you multiply each of them by a scalar, and you add together all the products thus obtained, then you obtain a linear combination.

Are all polynomials a vector space?

The set of all polynomials with real coefficients is a real vector space, with the usual oper- ations of addition of polynomials and multiplication of polynomials by scalars (in which all coefficients of the polynomial are multiplied by the same real number). number in Q( √ 2).

How do you show that a subset of a vector space is a subspace?

To check that a subset U of V is a subspace, it suffices to check only a few of the conditions of a vector space….Then U is a subspace of V if and only if the following three conditions hold.

1. additive identity: 0∈U;
2. closure under addition: u,v∈U⇒u+v∈U;
3. closure under scalar multiplication: a∈F, u∈U⟹au∈U.

Is a basis a linear combination?

In mathematics, a set B of vectors in a vector space V is called a basis if every element of V may be written in a unique way as a finite linear combination of elements of B. Equivalently, a set B is a basis if its elements are linearly independent and every element of V is a linear combination of elements of B.

How do you determine if a matrix is a linear combination of other matrices?

In other words, if you take a set of matrices, you multiply each of them by a scalar, and you add together all the products thus obtained, then you obtain a linear combination. Note that all the matrices involved in a linear combination need to have the same dimension (otherwise matrix addition would not be possible).