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In each case, construct a matrix with the requisite properties or explain why no such
Chapter 3, Problem 8(choose chapter or problem)
In each case, construct a matrix with the requisite properties or explain why no such matrix exists.
a. The column space has basis \(\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right]\), and the nullspace contains \(\left[\begin{array}{l} 1 \\ 2 \\ 0 \end{array}\right]\).
b. The nullspace contains \(\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right]\), \(\left[\begin{array}{r} -1 \\ 2 \\ 1 \end{array}\right]\), and the row space contains \(\left[\begin{array}{r} 1 \\ 1 \\ -1 \end{array}\right]\).
c. The column space has basis \(\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right]\), \(\left[\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right]\), and the row space has basis \(\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]\), \(\left[\begin{array}{l} 2 \\ 0 \\ 1 \end{array}\right]\).
d. The column space and the nullspace both have basis \(\left[\begin{array}{l} 1 \\ 0 \end{array}\right]\).
e. The column space and the nullspace both have basis \(\left[\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right]\)
Questions & Answers
(4 Reviews)
QUESTION:
In each case, construct a matrix with the requisite properties or explain why no such matrix exists.
a. The column space has basis \(\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right]\), and the nullspace contains \(\left[\begin{array}{l} 1 \\ 2 \\ 0 \end{array}\right]\).
b. The nullspace contains \(\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right]\), \(\left[\begin{array}{r} -1 \\ 2 \\ 1 \end{array}\right]\), and the row space contains \(\left[\begin{array}{r} 1 \\ 1 \\ -1 \end{array}\right]\).
c. The column space has basis \(\left[\begin{array}{l} 1 \\ 0 \\ 1 \end{array}\right]\), \(\left[\begin{array}{l} 0 \\ 1 \\ 1 \end{array}\right]\), and the row space has basis \(\left[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}\right]\), \(\left[\begin{array}{l} 2 \\ 0 \\ 1 \end{array}\right]\).
d. The column space and the nullspace both have basis \(\left[\begin{array}{l} 1 \\ 0 \end{array}\right]\).
e. The column space and the nullspace both have basis \(\left[\begin{array}{l} 1 \\ 0 \\ 0 \end{array}\right]\)
ANSWER:Step 1 of 9
Row Space, also known as a span of rows, is the collection of all linear combinations of its rows. Column space is the sum of all possible linear arrangements of a given set of columns, or span of columns. Null space is the collection of all homogeneous equation solutions.
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Review this written solution for 965464) viewed: 342 isbn: 9781429215213 | Linear Algebra: A Geometric Approach - 2 Edition - Chapter 3.4 - Problem 8
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Review this written solution for 965464) viewed: 342 isbn: 9781429215213 | Linear Algebra: A Geometric Approach - 2 Edition - Chapter 3.4 - Problem 8
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