16 Questions
What is the column space of a matrix?
The set of all linear combinations of the columns of the matrix
When does the equation Ax = b have a solution?
When b can be written as a linear combination of the columns of A
What does it mean for a matrix A to be invertible?
N (A) = {0}
In the context of matrix A, what does N (A) represent?
The null space containing vectors from Rn
What is the relationship between C(B) and N (A), if B is the nullspace matrix of A?
C(B) = N (A)
Which set contains all column vectors of length n?
$Rn$
What is the vector space that consists of all column vectors of length 3?
$R3$
In vector spaces, what can be done with two vectors in $V$?
$V$ allows addition and scalar multiplication
What does it mean for C(A) to be closed under linear combinations?
If a, b are in C(A), then ua + vb = u(Ax) + v(Ay) = A(ux + vy) = Aw
What must be true for a matrix A to be invertible?
N (A) = {0}
What is the general solution of Ax = b, where A = $\begin{bmatrix} 1 & 2 & 3 & 5 \ 2 & 4 & 8 & 12 \end{bmatrix}$ and b = $\begin{bmatrix} 1 \ 0 \ 5 \end{bmatrix}$?
$x = t\begin{bmatrix} -2 \ 4 \end{bmatrix} + u\begin{bmatrix} 4 \ -2 \end{bmatrix} + w\begin{bmatrix} -1 \ 0 \end{bmatrix}$
What is a particular solution of Ax = b, where A = $\begin{bmatrix} 1 & 2 & 3 & 5 \ 2 & 4 & 8 & 12 \end{bmatrix}$ and b = $\begin{bmatrix} 1 \ 0 \ 5 \end{bmatrix}$?
$\begin{bmatrix} 4 \ 0 \ -1 \ 0 \end{bmatrix}$
How can the general solution of Ax = b be obtained from Rx = d?
By finding the solutions of Rx = 0 and solving Rx = d for pivot variables
What is the null space of A, where A = $\begin{bmatrix} 1 & 2 & 3 & 5 \ 2 & 4 & 8 & 12 \end{bmatrix}$?
$N(A) = span\left(\left{\begin{bmatrix} -2 \ 1 \ 0 \ 0 \end{bmatrix}, \begin{bmatrix} -2 \ 0 \ -1 \ 1 \end{bmatrix}\right}\right)$
What is the particular solution of Rx = $\begin{bmatrix} 4 \ -1 \ 0 \end{bmatrix}$, given that R = $\begin{bmatrix} 0 & 0 & 1 & 1 \ 3 & 6 & 7 & 13 \ 0 & 0 & 0 & 0 \end{bmatrix}$?
$xParticular = \begin{bmatrix} t \ u \ v \ w \end{bmatrix}= \begin{bmatrix} t+3u+v+u/2-w/2-1/2 \ u \ v \ w \end{bmatrix}$
What does the complete set of solutions to Ax = b include?
$xNullSpace + xParticular$
This quiz covers the topic of solving linear equations with special solutions using examples and specific mathematical operations. It includes methods for solving Ax = b, Ux = c, and Rx = d equations. The quiz also addresses how to reduce solving Ax = b to solving Rx = d.
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