Subspaces

Author:Ku, Yin Bon (Albert)

What is a subspace?

Simply speaking, a subset

of a vector space

is a subspace if

is a vector space itself when inheriting the addition and scalar multiplication defined in

. Definition:

is a subspace of

is a non-empty subset of

that satisfies the following conditions:

For any and in , is in .
For any in and real number , is in . (Note: In particular, we can set and it implies that the zero vector is in .)

Example 1:

, the set containing only the zero vector, is a subspace of any vector space. It is called the zero subspace. Example 2: For any non-negative integer

, the vector space of all polynomial of degree

with real coefficients, is a subspace of

, the vector space of all polynomials with real coefficients. Example 3: Any plane in

containing the origin is a subspace of

. For example,

is a subspace of

because

Zero vector is obviously in so is non-empty.
For any , and . Then and . Therefore, is also in .
For any real number and in i.e. . . Hence , which means is also in .

Example 4: Let

be the vector space of all real number sequences and

. Then

is a subspace of

because

The sequence of all zeros is in so is non-empty.
If are in , then for all even . Therefore, for all even and is also in .
For any real number and any in , for all even . Therefore, for all even and is also in .

Example 5: Let

be the vector space of all n x n matrices. Let

. Then

is a subspace of

because

Zero matrix is in so is non-empty.
For any two n x n matrices in , they are upper triangular. Hence is obviously upper triangular and thus in .
For any real number and any n x n matrix in , is upper triangular. Therefore, is also upper triangular and thus in .

Exercise

Check the box if the statement is true.

Select all that apply

A
Any line through the origin in is a subspace of .
B
The set of all continuous functions from to is a subspace of all functions from to .
C
The set of all non-invertible n x n matrices is a subspace of .
D
The union of two non-parallel lines through the origin in is a subspace of .
E
The set of all 3 x 3 matrices whose entries in the first column are zero is not a subspace of .
F
The set is a subspace of .
G
The set of all vectors in such that their length are less than 1 is not a subspace of .

Check my answer (3)

Subspaces

What is a subspace?

Exercise

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