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Chapter 4: Q4.5-28Q (page 191)
Show that the space \(C\left( \mathbb{R} \right)\) of all continuous functions defined on the real line is infinite-dimensional.
It is proved that the space \(C\left( \mathbb{R} \right)\) of all continuous functions is infinite-dimensional.
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In Exercises 27-30, use coordinate vectors to test the linear independence of the sets of polynomials. Explain your work
\({\left( {{\bf{1}} - t} \right)^{\bf{2}}}\),\(t - {\bf{2}}{t^{\bf{2}}} + {t^{\bf{3}}}\),\({\left( {{\bf{1}} - t} \right)^{\bf{3}}}\)
Question 3: On any given day, a student is either healthy or ill. Of the students who are healthy today, 95% will be healthy tomorrow. Of the students who are ill today, 55% will still be ill tomorrow.
a. What is the stochastic matrix for this situation?
b. Suppose 20% of the students are ill on Monday. What fraction or percentage of the students are likely to be ill on Tuesday? On Wednesday?
c. If a student is well today, what is the probability that he or she will be well two days from now?
Question: In Exercises 25 and 26, A denotes a \(m \times n\) matrix. Mark each statement True or False. Justify each answer.
26.
a. A null space is a vector space.
b. The column space of a \(m \times n\) matrix is in \({\mathbb{R}^m}\).
c. Col A is the set of all solutions of \(A{\mathop{\rm x}\nolimits} = b\).
d. Nul A is the kernel of the mapping \({\mathop{\rm x}\nolimits} \mapsto A{\mathop{\rm x}\nolimits} \).
e. The range of a linear transformation is a vector space.
f. The set of all solutions of a homogeneous linear differential equation is the kernel of a linear transformation.
Let \(B = \left\{ {{{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}} \right\}\) be a basis for a vector space \(V\). Explain why the \(B - \)coordinate vectors of \({{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}\) are the columns \({{\mathop{\rm e}\nolimits} _1},...,{{\mathop{\rm e}\nolimits} _n}\) of the \(n \times n\) identity matrix.
Answer:
The \(B - \)coordinate vectors of \({{\mathop{\rm b}\nolimits} _1},...,{{\mathop{\rm b}\nolimits} _n}\) are columns \({{\mathop{\rm e}\nolimits} _1},...,{{\mathop{\rm e}\nolimits} _n}\) of the \(n \times n\) identity matrix.
A homogeneous system of twelve linear equations in eight unknowns has two fixed solutions that are not multiples of each other, and all other solutions are linear combinations of these two solutions. Can the set of all solutions be described with fewer than twelve homogeneous linear equations? If so, how many? Discuss.
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