# OEF subspace definition --- Introduction ---

This module actually contains 21 exercises on the definition of subspaces of vector spaces. You are given a vector space and a subset defined in various ways; up to you to determine whether the subset is a subspace.

See also the collections of exercises on vector spaces in general or defition of vector spaces.

### Continuous functions

Let be the -vector space of real functions

,

and let be the subset of composed of the function 0 and functions which are on [,].

Is a subspace of ?

### Increasing functions

Let be the -vector space of real functions

,

and let be the subset of composed of the function 0 and functions which are on [,].

Is a subspace of ?

### Crossed Matrices

Let be the vector space of matrices of size , and let be the subset of composed of matrices such that

Is a subspace of ?

### Matrices and determinant

Let be the vector space of matrices of size ×, and let be the subset of composed of matrices M whose is equal to .

Is a subspace of ?

### Matrices and elements

Let be the vector space of matrices of size ×, and let be the subset of composed of matrices M such that .

Is a subspace of ?

### Multiplied matrices

Let be the vector space of matrices of size , and let be the subset of composed of matrices
.

Is a subspace of ?

### Matrices and rank

Let be the vector space of matrices of size ×, and let be the subset of composed of matrices of rank .

Is a subspace of ?

You must give all the good replies.

### Matrices with power

Let be the vector space of matrices of size ×, and let be the subset of composed of matrices M such that M=0.

Is a subspace of ?

### Periodic functions

Let be the vector space of real continuous functions over , and let be the subset of composed of the function 0 and periodic functions whose period .

Is a subspace of ?

### Polynomials and coefficients

Let be the -vector space of real polynomials P(X) of degree less than or equal to , and let be the subset of composed of polynomials such that the of its coefficients equals .

Is a subspace of ?

### Polynomials and degrees

Let be the vector space of polynomials over , and let be the subset of composed of polynomials with degree .

Is a subspace of ?

### Polynomials and integral

Let be the vector space over of polynomials, and let be the subset of composed of polynomials such that

Is a subspace of ?

### Polynomials and integral II

Let be the vector space over of polynomials, and let be the subset of composed of polynomials such that

Is a subspace of ?

### Polynomials and roots

Let be the -vector space of real polynomials P(X) of degree less than or equal to , and let be the subset of composed of polynomials such that the of its roots (real or complex, counted with multiplicity) equals 0.

Is a subspace of ?

### Polynomials and roots II

Let be the -vector space of real polynomials P(X) of degree less than or equal to , and let be the subset of composed of polynomials with .

Is a subspace of ?

### Polynomials and values

Let be the vector space of polynomials over , and let be the subset of composed of polynomials such that .

Is a subspace of ?

### Polynomials and values II

Let be the vector space of polynomials over , and let be the subset of composed of polynomials (X) such that .

Is a subspace of ?

### Polynomials and values III

Let = [X] be the vector space of polynomials over , and let be the subset of composed of polynomials such that .

Is a subspace of ?

### Real functions

Let be the -vector space of real functions

,

and let be the subset of composed of functions (x) .

Is a subspace of ?

### Square matrices

Let be the vector space of matrices of size ×, and let be the subset of composed of matrices.

Is a subspace of ?

### Vectors of R^3

Let be the vector space of dimension 3 over , and let be the subset of composed of vectors such that .

Is a subspace of ?

Other exercises on: vector spaces   linear algebra