MMiDS 2.2: Self-Assessment Quiz

What is the span of the vectors \( \mathbf{w}_1, \mathbf{w}_2, \ldots, \mathbf{w}_m \in \mathbb{R}^n \)?

• a) The set of all linear combinations of the \( \mathbf{w}_j \)'s.
• b) The set of all vectors orthogonal to the \( \mathbf{w}_j \)'s.
• c) The set of all scalar multiples of the \( \mathbf{w}_j \)'s.
• d) The empty set.

Which of the following is a necessary condition for a subset \( U \) of \( \mathbb{R}^n \) to be a linear subspace?

• a) \( U \) must contain only non-zero vectors.
• b) \( U \) must be closed under vector addition and scalar multiplication.
• c) \( U \) must be a finite set.
• d) \( U \) must be a proper subset of \( \mathbb{R}^n \).

If \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_m \) are linearly independent vectors, which of the following is true?

• a) There exist scalars \( \alpha_1, \alpha_2, \ldots, \alpha_m \) such that \( \sum_{i=1}^m \alpha_i \mathbf{v}_i = 0 \) with at least one \( \alpha_i \neq 0 \).
• b) There do not exist scalars \( \alpha_1, \alpha_2, \ldots, \alpha_m \) such that \( \sum_{i=1}^m \alpha_i \mathbf{v}_i = 0 \) unless \( \alpha_i = 0 \) for all \( i \).
• c) The vectors \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_m \) span \( \mathbb{R}^n \) for any \( n \).
• d) The vectors \( \mathbf{v}_1, \mathbf{v}_2, \ldots, \mathbf{v}_m \) are orthogonal.

Which of the following matrices has full column rank?

• a) \( \begin{pmatrix} 1 & 2 \\ 2 & 4 \end{pmatrix} \)
• b) \( \begin{pmatrix} 1 & 0 \\ 1 & 1 \end{pmatrix} \)
• c) \( \begin{pmatrix} 1 & 1 \\ 1 & 1 \end{pmatrix} \)
• d) \( \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 9 \\ 7 & 8 & 15 \end{pmatrix} \)

Which of the following statements about bases is FALSE?

• a) A basis of a linear subspace spans the subspace.
• b) A basis of a linear subspace consists of linearly independent vectors.
• c) A basis of a linear subspace is unique.
• d) Every non-zero linear subspace has a basis.

What is the dimension of a linear subspace?

• a) The number of vectors in any spanning set of the subspace.
• b) The number of linearly independent vectors in the subspace.
• c) The number of vectors in any basis of the subspace.
• d) The number of orthogonal vectors in the subspace.

Let \( u_1, \ldots, u_m \) be an orthonormal list. Which of the following is true?

• a) \( \langle u_i, u_j \rangle = 1 \) for all \( i,j \).
• b) \( \langle u_i, u_j \rangle = 0 \) for all \( i \neq j \) and \( \|u_i\| = 1 \) for all \( i \).
• c) \( \langle u_i, u_j \rangle = 1 \) for all \( i \neq j \) and \( \|u_i\| = 0 \) for all \( i \).
• d) None of the above.

Let \( q_1, \ldots, q_m \) be an orthonormal basis of a subspace \( U \) and let \( w \in U \). What is the coefficient of \( q_j \) in the orthonormal expansion of \( w \)?

• a) \( \langle w, q_j \rangle \)
• b) \( \|w\| \|q_j\| \)
• c) \( \frac{w_j}{q_j} \)
• d) None of the above.

The Gram-Schmidt process takes as input:

• a) An orthonormal list and outputs a basis.
• b) A basis and outputs an orthonormal list.
• c) An orthonormal list and outputs another orthonormal list.
• d) A basis and outputs another basis.

A square matrix \( A \) is nonsingular if and only if:

• a) Its columns are linearly independent.
• b) It has full rank.
• c) It has an inverse.
• d) All of the above.