Master Subspaces in Linear Algebra: Definition, Properties, and Examples

Question 1

Definition of a Subspace: Which of the following accurately defines a subspace of a vector space?

Accepted Answer

A subset that remains closed under vector addition and scalar multiplication

Answer

A subset that contains the zero vector

Answer

A subset that spans the entire vector space

Answer

A subset that is linearly independent

Question 2

Non-Property of Subspaces: Which characteristic is not a property of subspaces?

Accepted Answer

Always being linearly dependent

Answer

Closed under vector addition

Answer

Containing the zero vector

Answer

Closed under scalar multiplication

Question 3

Determining Subspace Membership: Consider the set of vectors {(1, 0, 0), (0, 1, 0), (0, 0, 1)}. Does this set form a subspace of R³?

Accepted Answer

Yes, it satisfies all subspace properties.

Answer

No, it is not closed under vector addition.

Answer

No, it does not contain the zero vector.

Answer

No, it is not closed under scalar multiplication.

Question 4

Basis for a Subspace: Which set of vectors forms a basis for the subspace spanned by {(1, 0, 1), (0, 1, 1), (1, 1, 0)}?

Accepted Answer

{(1, 0, 1), (0, 1, 1)}

Answer

{(1, 0, 1), (0, 1, 1), (1, 1, 0)}

Answer

{(0, 1, 1), (1, 1, 0)}

Answer

{(1, 0, 1), (1, 1, 0), (0, 1, 1), (0, 0, 1)}

Question 5

Subspace of Polynomial Space: Which of the following is a subspace of the vector space of polynomials of degree at most 2?

Accepted Answer

Polynomials with constant term equal to 0

Answer

Polynomials with leading coefficient equal to 1

Answer

Polynomials with roots at x = 1 and x = 2

Answer

Polynomials with odd degree

Question 6

Which property does NOT belong to a subspace?

Accepted Answer

Spans the entire vector space

Answer

Closed under vector addition

Answer

Contains the zero vector

Answer

Closed under scalar multiplication

Question 7

Consider the linear transformation T: R³ → R² defined by T(x, y, z) = (x + y, y + z). Determine if the image of T is a subspace of R².

Accepted Answer

Yes, because the image of T is closed under vector addition and scalar multiplication, and it contains the zero vector.

Answer

No, because the image of T is not closed under vector addition.

Answer

No, because the zero vector is not in the image of T.

Answer

No, because the image of T is not a vector space.

Question 8

Which of the following is not a defining property of a subspace?

Accepted Answer

Contains all linear combinations of its vectors

Answer

Contains the zero vector

Answer

Closed under vector addition

Answer

Closed under scalar multiplication

Question 9

What is the dimension of the subspace of R^n spanned by the vectors {(1, 0, ..., 0), (0, 1, ..., 0), ..., (0, ..., 0, 1)}?

Accepted Answer

n

Answer

n-1

Answer

1

Answer

0

Question 10

Determine if the set of all symmetric matrices in R^(n x n) forms a subspace.

Accepted Answer

Yes, it is a subspace

Answer

No, it does not contain the zero matrix

Answer

No, it is not closed under matrix addition

Answer

No, it is not closed under scalar multiplication

Question 11

State whether the following statement is true or false: Every vector space is a subspace of itself.

Accepted Answer

True

Answer

False

Question 12

Which of the following is an essential characteristic of a subspace?

Accepted Answer

It contains the zero vector and is closed under vector addition.

Answer

It is a subset of a vector space and is closed under scalar multiplication.

Answer

It is linearly independent and spans the entire vector space.

Answer

It has the same dimension as the vector space it belongs to.

Question 13

Consider the set of all vectors in R^3 with zero as the second component. Does this set form a subspace of R^3?

Accepted Answer

Yes

Answer

It depends on the specific values of the first and third components.

Answer

No, because it does not contain the zero vector.

Answer

Yes, but only if the other components are non-zero.

Question 14

Let S be the set of all vectors (a, b, c) in R^3 such that 2a + 3b - c = 0. Is S a subspace of R^3?

Accepted Answer

Yes

Answer

No, because it does not contain all vectors in R^3.

Answer

Yes, but only if a, b, and c are all non-zero.

Answer

It depends on the specific values of a, b, and c.

Question 15

Which of the following statements is equivalent to the definition of a subspace?

Accepted Answer

Any linear combination of vectors in the subspace also belongs to the subspace.

Answer

The subspace is a subset of the vector space with the same dimension.

Answer

The subspace contains the zero vector and is closed under vector addition.

Question 16

Let U and W be subspaces of a vector space V. Which of the following is always true?

Accepted Answer

The intersection of U and W is also a subspace of V.

Answer

The union of U and W is also a subspace of V.

Answer

The sum of the dimensions of U and W is equal to the dimension of V.

Question 17

Which of the following is NOT a defining property of a subspace?

Accepted Answer

Contains only the zero vector

Answer

Closed under vector addition

Answer

Closed under scalar multiplication

Answer

Contains the zero vector

Question 18

Which of the following is equivalent to the statement 'S is a subspace of V'?

Accepted Answer

0 ∈ S and S is closed under vector addition and scalar multiplication

Answer

0 ∉ S and S is closed under vector addition

Answer

0 ∈ S and S spans V

Question 19

Let V be the vector space of all 2x2 matrices over the field of real numbers. Is the set of all matrices with trace 0 a subspace of V?

Accepted Answer

Yes

Answer

No, it is not closed under scalar multiplication

Answer

No, it is not closed under vector addition

Answer

Cannot be determined

Question 20

If a subspace W of V has dimension k, then the dimension of the quotient space V/W is

Accepted Answer

dim(V) - k

Answer

dim(W)

Answer

k

Answer

dim(V) + k

Question 21

What is the subspace spanned by the empty set of vectors?

Accepted Answer

The zero subspace

Answer

A subspace of dimension -1

Answer

Isomorphic to the subspace spanned by the unit vector

Answer

The entire vector space

Question 22

Consider a vector space V over a field F. Which of the following sets of vectors in V is NOT a subspace of V?

Accepted Answer

The set of all vectors in V with a rational first component

Answer

The set of all vectors in V that are multiples of a fixed vector

Answer

The set of all vectors in V with zero third component

Question 23

In the context of subspaces, what is the relationship between two subspaces W and another subspace of a vector space V?

Accepted Answer

The intersection of W and any other subspace of V is also a subspace of V

Answer

The difference between W and any other subspace of V is also a subspace of V

Answer

The union of W and any other subspace of V is also a subspace of V

Question 24

Consider the subspace W of R^3 spanned by the vectors (1, 0, 1) and (0, 1, 1). Which vector is NOT a member of W?

Accepted Answer

(1, 1, 0)

Answer

(0, 2, 2)

Answer

(3, 1, 4)

Answer

(2, 0, 2)

Question 25

In a finite-dimensional vector space, what is the relationship between the dimensions of two subspaces and their intersection?

Accepted Answer

The sum of their dimensions is greater than or equal to the dimension of their intersection

Answer

The sum of their dimensions is always less than the dimension of their intersection

Answer

The sum of their dimensions is always equal to the dimension of their intersection

Question 26

Consider the subspace W of R^4 spanned by the vectors (1, 0, 1, 0), (0, 1, 0, 1), and (1, 1, 1, 1). Which set of vectors forms a basis for W?

Accepted Answer

{(1, 0, 1, 0), (0, 1, 0, 1)}

Answer

{(0, 1, 0, 1), (1, 1, 1, 1)}

Answer

{(1, 1, 1, 1), (1, 0, 1, 0)}

Answer

{(1, 0, 1, 0), (0, 1, 0, 1), (1, 1, 1, 1)}

Question 27

Regarding a subspace W of a vector space V, what fundamental characteristic must hold true?

Accepted Answer

W and V have the same zero vector

Answer

W and V have the same number of linearly independent vectors

Answer

W and V have the same dimension

Answer

W and V have the same set of basis vectors

Question 28

Is the set S = {(x, y) | x + y = 1} a subspace of R²?

Accepted Answer

No, because S does not contain the zero vector (0, 0).

Answer

Yes, because S is closed under vector addition.

Answer

Yes, because S is a line in R².

Answer

Yes, because S is closed under scalar multiplication.

Question 29

Consider the set of all 2x2 matrices with a trace of 0. Is this set a subspace of the vector space of all 2x2 matrices?

Accepted Answer

Yes, because it is closed under both addition and scalar multiplication.

Answer

No, because it is not closed under addition.

Answer

No, because it does not contain the zero matrix.

Question 30

Let W be the set of all polynomials of degree less than or equal to 3 in P₃. Is W a subspace of P₃?

Accepted Answer

Yes, because it satisfies all the conditions of a subspace: it contains the zero polynomial, is closed under addition, and is closed under scalar multiplication.

Answer

No, because it does not contain all polynomials in P₃.

Answer

No, because it is not closed under addition.