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For each of the following matrices A, give bases for R(A), N(A), C(A), and N(AT). Check
Chapter 3, Problem 3(choose chapter or problem)
For the following matrices A, give a basis for \(\mathbf{R}(A), \mathbf{N}(A), \mathbf{C}(A)\), and \(\mathbf{N}\left(A^{\top}\right)\). Check dimensions and orthogonality.
a. \(A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 2 & 4 & 6 \end{array}\right]\)
b. \(A=\left[\begin{array}{lll} 2 & 1 & 3 \\ 4 & 3 & 5 \\ 3 & 3 & 3 \end{array}\right]\)
c. \(A=\left[\begin{array}{rrrr} 1 & -2 & 1 & 0 \\ 2 & -4 & 3 & -1 \end{array}\right]\)
d. \(A=\left[\begin{array}{rrrrr} 1 & -1 & 1 & 1 & 0 \\ 1 & 0 & 2 & 1 & 1 \\ 0 & 2 & 2 & 2 & 0 \\ -1 & 1 & -1 & 0 & -1 \end{array}\right]\)
e. \(A=\left[\begin{array}{rrrrr} 1 & 1 & 0 & 1 & -1 \\ 1 & 1 & 2 & -1 & 1 \\ 2 & 2 & 2 & 0 & 0 \\ -1 & -1 & 2 & -3 & 3 \end{array}\right]\)
f. \(A=\left[\begin{array}{rrrrrr} 1 & 1 & 0 & 5 & 0 & -1 \\ 0 & 1 & 1 & 3 & -2 & 0 \\ -1 & 2 & 3 & 4 & 1 & -6 \\ 0 & 4 & 4 & 12 & -1 & -7 \end{array}\right]\)
Questions & Answers
(4 Reviews)
QUESTION:
For the following matrices A, give a basis for \(\mathbf{R}(A), \mathbf{N}(A), \mathbf{C}(A)\), and \(\mathbf{N}\left(A^{\top}\right)\). Check dimensions and orthogonality.
a. \(A=\left[\begin{array}{lll} 1 & 2 & 3 \\ 2 & 4 & 6 \end{array}\right]\)
b. \(A=\left[\begin{array}{lll} 2 & 1 & 3 \\ 4 & 3 & 5 \\ 3 & 3 & 3 \end{array}\right]\)
c. \(A=\left[\begin{array}{rrrr} 1 & -2 & 1 & 0 \\ 2 & -4 & 3 & -1 \end{array}\right]\)
d. \(A=\left[\begin{array}{rrrrr} 1 & -1 & 1 & 1 & 0 \\ 1 & 0 & 2 & 1 & 1 \\ 0 & 2 & 2 & 2 & 0 \\ -1 & 1 & -1 & 0 & -1 \end{array}\right]\)
e. \(A=\left[\begin{array}{rrrrr} 1 & 1 & 0 & 1 & -1 \\ 1 & 1 & 2 & -1 & 1 \\ 2 & 2 & 2 & 0 & 0 \\ -1 & -1 & 2 & -3 & 3 \end{array}\right]\)
f. \(A=\left[\begin{array}{rrrrrr} 1 & 1 & 0 & 5 & 0 & -1 \\ 0 & 1 & 1 & 3 & -2 & 0 \\ -1 & 2 & 3 & 4 & 1 & -6 \\ 0 & 4 & 4 & 12 & -1 & -7 \end{array}\right]\)
ANSWER:Step 1 of 25
A square matrix with actual numbers or components is said to be orthogonal if its transpose corresponds to the inverse matrices. Alternatively, we can say that a matrix of squares is orthogonal if the identity matrix emerges when the square matrix is multiplied by its transpose.
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