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Chapter 4: Q21Q (page 191)

The first four Hermite polynomials are \(1,2t, - 2 + 4{t^2},\) and \( - 12t + 8{t^3}\). These polynomials arise naturally in the study of certain important differential equations in mathematical physics. Show that the first four Hermite polynomials form a basis of \({{\mathop{\rm P}\nolimits} _3}\).

Short Answer

Expert verified

It is proved that the first four Hermite polynomials form a basis of \({{\mathop{\rm P}\nolimits} _3}\).

Step by step solution

01

Basis theorem

Theorem 12states that let\(V\)be a p-dimensional vector space;\(p \ge 1\), then anylinearly independent set of exactly \(p\) elements in \(V\) is automatically a basis for \(V\). Any set of exactly \(p\) elements that span \(V\) is automatically a basis for \(V\).

02

Show that the first four Hermite polynomials form a basis of \({{\mathop{\rm P}\nolimits} _3}\)

The columns of the matrix are the coordinate vectors of the Hermite polynomials corresponding to the standard basis\(\left\{ {1,t,{t^2},{t^3}} \right\}\)of\({{\mathop{\rm P}\nolimits} _3}\). Thus,

\(A = \left[ {\begin{array}{*{20}{c}}1&0&{ - 2}&0\\0&2&0&{ - 12}\\0&0&4&0\\0&0&0&8\end{array}} \right]\)

There are four pivot columns in the matrix; so its columns are linearly independent. The Hermite polynomials themselves are linearly independent in\({{\mathop{\rm P}\nolimits} _3}\)because the coordinate vectors of Hermite polynomials form a linearly independent set. The basis theorem asserts that the Hermite polynomials form a basis for\({{\mathop{\rm P}\nolimits} _3}\)because there are four Hermite polynomials and\(\dim {{\mathop{\rm P}\nolimits} _3} = 4\).

Thus, it is proved that the first four Hermite polynomials form a basis of \({{\mathop{\rm P}\nolimits} _3}\).

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Most popular questions from this chapter

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?

In Exercise 17, Ais an \(m \times n\] matrix. Mark each statement True or False. Justify each answer.

17. a. The row space of A is the same as the column space of \({A^T}\].

b. If B is any echelon form of A, and if B has three nonzero rows, then the first three rows of A form a basis for Row A.

c. The dimensions of the row space and the column space of A are the same, even if Ais not square.

d. The sum of the dimensions of the row space and the null space of A equals the number of rows in A.

e. On a computer, row operations can change the apparent rank of a matrix.

Suppose a nonhomogeneous system of six linear equations in eight unknowns has a solution, with two free variables. Is it possible to change some constants on the equations鈥� right sides to make the new system inconsistent? Explain.

Question: Determine if the matrix pairs in Exercises 19-22 are controllable.

19. \(A = \left( {\begin{array}{*{20}{c}}{.9}&1&0\\0&{ - .9}&0\\0&0&{.5}\end{array}} \right),B = \left( {\begin{array}{*{20}{c}}0\\1\\1\end{array}} \right)\).

Let \({M_{2 \times 2}}\) be the vector space of all \(2 \times 2\) matrices, and define \(T:{M_{2 \times 2}} \to {M_{2 \times 2}}\) by \(T\left( A \right) = A + {A^T}\), where \(A = \left( {\begin{array}{*{20}{c}}a&b\\c&d\end{array}} \right)\).

  1. Show that \(T\)is a linear transformation.
  2. Let \(B\) be any element of \({M_{2 \times 2}}\) such that \({B^T} = B\). Find an \(A\) in \({M_{2 \times 2}}\) such that \(T\left( A \right) = B\).
  3. Show that the range of \(T\) is the set of \(B\) in \({M_{2 \times 2}}\) with the property that \({B^T} = B\).
  4. Describe the kernel of \(T\).
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