Solving Linear Equations with Special Solutions
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Questions and Answers

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 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?

  • 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?

    <p>The null space containing vectors from Rn</p> Signup and view all the answers

    What is the relationship between C(B) and N (A), if B is the nullspace matrix of A?

    <p>C(B) = N (A)</p> Signup and view all the answers

    Which set contains all column vectors of length n?

    <p>$Rn$</p> Signup and view all the answers

    What is the vector space that consists of all column vectors of length 3?

    <p>$R3$</p> Signup and view all the answers

    In vector spaces, what can be done with two vectors in $V$?

    <p>$V$ allows addition and scalar multiplication</p> Signup and view all the answers

    What does it mean for C(A) to be closed under linear combinations?

    <p>If a, b are in C(A), then ua + vb = u(Ax) + v(Ay) = A(ux + vy) = Aw</p> Signup and view all the answers

    What must be true for a matrix A to be invertible?

    <p>N (A) = {0}</p> Signup and view all the answers

    What is the general solution of Ax = b, where A = $\begin{bmatrix} 1 & 2 & 3 & 5 \ 2 & 4 & 8 & 12 \end{bmatrix}$ and b = $\begin{bmatrix} 1 \ 0 \ 5 \end{bmatrix}$?

    <p>$x = t\begin{bmatrix} -2 \ 4 \end{bmatrix} + u\begin{bmatrix} 4 \ -2 \end{bmatrix} + w\begin{bmatrix} -1 \ 0 \end{bmatrix}$</p> Signup and view all the answers

    What is a particular solution of Ax = b, where A = $\begin{bmatrix} 1 & 2 & 3 & 5 \ 2 & 4 & 8 & 12 \end{bmatrix}$ and b = $\begin{bmatrix} 1 \ 0 \ 5 \end{bmatrix}$?

    <p>$\begin{bmatrix} 4 \ 0 \ -1 \ 0 \end{bmatrix}$</p> Signup and view all the answers

    How can the general solution of Ax = b be obtained from Rx = d?

    <p>By finding the solutions of Rx = 0 and solving Rx = d for pivot variables</p> Signup and view all the answers

    What is the null space of A, where A = $\begin{bmatrix} 1 & 2 & 3 & 5 \ 2 & 4 & 8 & 12 \end{bmatrix}$?

    <p>$N(A) = span\left(\left{\begin{bmatrix} -2 \ 1 \ 0 \ 0 \end{bmatrix}, \begin{bmatrix} -2 \ 0 \ -1 \ 1 \end{bmatrix}\right}\right)$</p> Signup and view all the answers

    What is the particular solution of Rx = $\begin{bmatrix} 4 \ -1 \ 0 \end{bmatrix}$, given that R = $\begin{bmatrix} 0 & 0 & 1 & 1 \ 3 & 6 & 7 & 13 \ 0 & 0 & 0 & 0 \end{bmatrix}$?

    <p>$xParticular = \begin{bmatrix} t \ u \ v \ w \end{bmatrix}= \begin{bmatrix} t+3u+v+u/2-w/2-1/2 \ u \ v \ w \end{bmatrix}$</p> Signup and view all the answers

    What does the complete set of solutions to Ax = b include?

    <p>$xNullSpace + xParticular$</p> Signup and view all the answers

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