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Subspaces

This quiz covers the concept of subspaces in linear algebra. It includes questions on the definition of subspaces, their properties, and examples of subspaces in different vector spaces.

15 Questions Published
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Questions

Question 1 Multiple Choice (Single Answer)

What is a subspace of a vector space?

  1. A non-empty subset of a vector space that is closed under vector addition and scalar multiplication
  2. A non-empty subset of a vector space that is closed under vector addition
  3. A non-empty subset of a vector space that is closed under scalar multiplication
  4. A non-empty subset of a vector space that is closed under vector addition and scalar multiplication and contains the zero vector
Question 2 Multiple Choice (Single Answer)

Which of the following is an example of a subspace of the vector space $\mathbb{R}^3$?

  1. The set of all vectors in $\mathbb{R}^3$ with zero as their first coordinate
  2. The set of all vectors in $\mathbb{R}^3$ with zero as their second coordinate
  3. The set of all vectors in $\mathbb{R}^3$ with zero as their third coordinate
  4. The set of all vectors in $\mathbb{R}^3$ with zero as their first and second coordinates
Question 3 Multiple Choice (Single Answer)

Which of the following is NOT a property of subspaces?

  1. They are closed under vector addition
  2. They are closed under scalar multiplication
  3. They contain the zero vector
  4. They are always finite-dimensional
Question 4 Multiple Choice (Single Answer)

If $\mathbf{u}$ and $\mathbf{v}$ are in a subspace $W$ of a vector space $V$, then $\mathbf{u} + \mathbf{v}$ is in $W$.

  1. True
  2. False
Question 5 Multiple Choice (Single Answer)

If $\mathbf{u}$ is in a subspace $W$ of a vector space $V$, and $c$ is a scalar, then $c\mathbf{u}$ is in $W$.

  1. True
  2. False
Question 6 Multiple Choice (Single Answer)

If $W_1$ and $W_2$ are subspaces of a vector space $V$, then their intersection $W_1 \cap W_2$ is also a subspace of $V$.

  1. True
  2. False
Question 7 Multiple Choice (Single Answer)

If $W_1$ and $W_2$ are subspaces of a vector space $V$, then their union $W_1 \cup W_2$ is also a subspace of $V$.

  1. True
  2. False
Question 8 Multiple Choice (Single Answer)

The set of all solutions to a system of linear equations is a subspace of the vector space of all vectors in the variables of the system.

  1. True
  2. False
Question 9 Multiple Choice (Single Answer)

The set of all polynomials of degree less than or equal to $n$ is a subspace of the vector space of all polynomials.

  1. True
  2. False
Question 10 Multiple Choice (Single Answer)

The set of all matrices of order $m \times n$ is a subspace of the vector space of all matrices.

  1. True
  2. False
Question 11 Multiple Choice (Single Answer)

Every subspace of a vector space is a vector space.

  1. True
  2. False
Question 12 Multiple Choice (Single Answer)

The dimension of a subspace of a vector space is always less than or equal to the dimension of the vector space.

  1. True
  2. False
Question 13 Multiple Choice (Single Answer)

The zero subspace of a vector space is the subspace consisting of the zero vector only.

  1. True
  2. False
Question 14 Multiple Choice (Single Answer)

The trivial subspace of a vector space is the subspace consisting of all vectors in the vector space.

  1. True
  2. False
Question 15 Multiple Choice (Single Answer)

The span of a set of vectors in a vector space is the smallest subspace of the vector space that contains the set of vectors.

  1. True
  2. False
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