Chapter 4: Q10E (page 191)
Question 10: Determine if \(P = \left[ {\begin{array}{*{20}{c}}1&{.2}\\0&{.8}\end{array}} \right]\) is a regular stochastic matrix.
Short Answer
\(P\)is not a regular stochastic matrix.
Chapter 4: Q10E (page 191)
Question 10: Determine if \(P = \left[ {\begin{array}{*{20}{c}}1&{.2}\\0&{.8}\end{array}} \right]\) is a regular stochastic matrix.
\(P\)is not a regular stochastic matrix.
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Get started for freeLet \(T:{\mathbb{R}^n} \to {\mathbb{R}^m}\) be a linear transformation.
a. What is the dimension of range of T if T is one-to-one mapping? Explain.
b. What is the dimension of the kernel of T (see section 4.2) if T maps \({\mathbb{R}^n}\) onto \({\mathbb{R}^m}\)? Explain.
Question 18: Suppose A is a \(4 \times 4\) matrix and B is a \(4 \times 2\) matrix, and let \({{\mathop{\rm u}\nolimits} _0},...,{{\mathop{\rm u}\nolimits} _3}\) represent a sequence of input vectors in \({\mathbb{R}^2}\).
Question: Exercises 12-17 develop properties of rank that are sometimes needed in applications. Assume the matrix \(A\) is \(m \times n\).
16. If \(A\) is an \(m \times n\) matrix of rank\(r\), then a rank factorization of \(A\) is an equation of the form \(A = CR\), where \(C\) is an \(m \times r\) matrix of rank\(r\) and \(R\) is an \(r \times n\) matrix of rank \(r\). Such a factorization always exists (Exercise 38 in Section 4.6). Given any two \(m \times n\) matrices \(A\) and \(B\), use rank factorizations of \(A\) and \(B\) to prove that rank\(\left( {A + B} \right) \le {\mathop{\rm rank}\nolimits} A + {\mathop{\rm rank}\nolimits} B\).
(Hint: Write \(A + B\) as the product of two partitioned matrices.)
Given \(T:V \to W\) as in Exercise 35, and given a subspace \(Z\) of \(W\), let \(U\) be the set of all \({\mathop{\rm x}\nolimits} \) in \(V\) such that \(T\left( {\mathop{\rm x}\nolimits} \right)\) is in \(Z\). Show that \(U\) is a subspace of \(V\).
Suppose \(A\) is \(m \times n\)and \(b\) is in \({\mathbb{R}^m}\). What has to be true about the two numbers rank \(\left[ {A\,\,\,{\rm{b}}} \right]\) and \({\rm{rank}}\,A\) in order for the equation \(Ax = b\) to be consistent?
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