Podcast
Questions and Answers
What is the column space of a matrix?
What is the column space of a matrix?
- The set of all vectors in N (A)
- The set of all solutions to the equation Ax = b
- The set of all linear combinations of the columns of the matrix (correct)
- The set of all vectors in Rn
When does the equation Ax = b have a solution?
When does the equation Ax = b have a solution?
- When A is a square matrix
- When b can be written as a linear combination of the columns of A (correct)
- When N (A) = {0}
- When C(A) = Rn
What does it mean for a matrix A to be invertible?
What does it mean for a matrix A to be invertible?
- N (A) and C(A) are not closed under linear combinations
- N (A) contains vectors from Rm
- C(A) = Rn
- N (A) = {0} (correct)
In the context of matrix A, what does N (A) represent?
In the context of matrix A, what does N (A) represent?
What is the relationship between C(B) and N (A), if B is the nullspace matrix of A?
What is the relationship between C(B) and N (A), if B is the nullspace matrix of A?
Which set contains all column vectors of length n?
Which set contains all column vectors of length n?
What is the vector space that consists of all column vectors of length 3?
What is the vector space that consists of all column vectors of length 3?
In vector spaces, what can be done with two vectors in $V$?
In vector spaces, what can be done with two vectors in $V$?
What does it mean for C(A) to be closed under linear combinations?
What does it mean for C(A) to be closed under linear combinations?
What must be true for a matrix A to be invertible?
What must be true for a matrix A to be invertible?
What is the general solution of Ax = b, where A = $\begin{bmatrix} 1 & 2 & 3 & 5 \ 2 & 4 & 8 & 12 \end{bmatrix}$ and b = $\begin{bmatrix} 1 \ 0 \ 5 \end{bmatrix}$?
What is the general solution of Ax = b, where A = $\begin{bmatrix} 1 & 2 & 3 & 5 \ 2 & 4 & 8 & 12 \end{bmatrix}$ and b = $\begin{bmatrix} 1 \ 0 \ 5 \end{bmatrix}$?
What is a particular solution of Ax = b, where A = $\begin{bmatrix} 1 & 2 & 3 & 5 \ 2 & 4 & 8 & 12 \end{bmatrix}$ and b = $\begin{bmatrix} 1 \ 0 \ 5 \end{bmatrix}$?
What is a particular solution of Ax = b, where A = $\begin{bmatrix} 1 & 2 & 3 & 5 \ 2 & 4 & 8 & 12 \end{bmatrix}$ and b = $\begin{bmatrix} 1 \ 0 \ 5 \end{bmatrix}$?
How can the general solution of Ax = b be obtained from Rx = d?
How can the general solution of Ax = b be obtained from Rx = d?
What is the null space of A, where A = $\begin{bmatrix} 1 & 2 & 3 & 5 \ 2 & 4 & 8 & 12 \end{bmatrix}$?
What is the null space of A, where A = $\begin{bmatrix} 1 & 2 & 3 & 5 \ 2 & 4 & 8 & 12 \end{bmatrix}$?
What is the particular solution of Rx = $\begin{bmatrix} 4 \ -1 \ 0 \end{bmatrix}$, given that R = $\begin{bmatrix} 0 & 0 & 1 & 1 \ 3 & 6 & 7 & 13 \ 0 & 0 & 0 & 0 \end{bmatrix}$?
What is the particular solution of Rx = $\begin{bmatrix} 4 \ -1 \ 0 \end{bmatrix}$, given that R = $\begin{bmatrix} 0 & 0 & 1 & 1 \ 3 & 6 & 7 & 13 \ 0 & 0 & 0 & 0 \end{bmatrix}$?
What does the complete set of solutions to Ax = b include?
What does the complete set of solutions to Ax = b include?