#
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
?

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- Description: collection of exercices on the definition of subspace of vector spaces. interactive exercises, online calculators and plotters, mathematical recreation and games
- Keywords: interactive mathematics, interactive math, server side interactivity, algebra, linear algebra, vector space, subspace, polynomial, matrix, function