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Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 4: Q51E (page 200)

The space $P_{11}$ is isomorphic to $R^{3 脳 4}$ .

Short Answer

Expert verified

The solution is that the statement is true.

Step by step solution

01

Determine the dimensions of the domain

Consider the space $P_{11}$ and $R^{3 脳 4}$ as follows.

The dimension of domain $P_{11}$ and the target space $R^{3 脳 4}$ are same.

02

Explanation of the solution

That is as follows.

$\dim (P_{11}) = \dim (R^{3 脳 4}) = 12$

Similarly as follows.

$f (x) = a + b x + c x^{2} + d x^{3} + e x^{4} + f x^{5} + g x^{6} + h x^{7} + i x^{8} + j x^{9} + k x^{10} + l x^{11}$

in $P_{11}$

Here, the monomial are as follows.

$1, x, x^{2}, x^{3}, x^{4}, x^{5}, x^{6}, x^{7}, x^{8}, x^{9}, x^{10}, x^{11}$

span $P_{11}$

So, these monomial are independent

$[\begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} \begin{matrix} a \\ b \\ c \end{matrix} \\ d \\ e \end{matrix} \\ f \\ g \end{matrix} \\ h \\ i \end{matrix} \\ j \\ k \end{matrix} \\ l \end{matrix}] in 鈥� 鈥� P_{11} 鈥� 鈥� 鈫� 鈥� [\begin{matrix} a & b & c & d \\ e & f & g & h \\ i & j & k & l \end{matrix}] in 鈥� 鈥� 鈥� R^{3 脳 4}$

The linear space $P_{11}$ and $R^{3 脳 4}$ have essentially the same structure.

Therefore, the space $P_{11}$ is isomorphic to $R^{3 脳 4}$ .

Thus, the given statement 鈥渢he space $P_{11}$ is isomorphic to $R^{3 脳 4}$ 鈥� is true.

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Most popular questions from this chapter

Find the kernel and nullity of the transformation $T i n T (M) = [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}] M - M [\begin{matrix} 1 & 2 \\ 0 & 1 \end{matrix}] f r o m R^{(2 脳 2)} t o R^{(2 脳 2)} .$ .

Find the transformation is linear and determine whether they are isomorphism.

Find the set of all polynomial $f (t)$ in $P_{2}$ such that $f (1) = 0$ , and determine its dimension.

Find the basis of all $2 脳 2$ matrix A such that $[\begin{matrix} 1 & 2 \\ 3 & 6 \end{matrix}] A = [\begin{matrix} 0 & 0 \\ 0 & 0 \end{matrix}]$ , and determine itsdimension.

Find the basis of all real linear space $C^{2}$ , and determine its dimension.

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