Q47E a.Find the change of basis matri... [FREE SOLUTION]

Chapter 4: Q47E (page 196)

a.Find the change of basis matrix S from the basis B considered in Exercise 28 to the standard basis $U = (1, t, t^{2})$ of $P_{2}$ considered in Exercise 27.
b.Verify the formula SB=AS for the matrices B and A you found in Exercises 28 and 27,respectively.
c.Find the change of basis matrix from U to B

Short Answer

Expert verified

a.The solution is $S = (\begin{matrix} 1 & 鈭� 1 & 1 \\ 0 & 1 & 鈭� 2 \\ 0 & 0 & 1 \end{matrix})$

b. Verified

c.The solution is $S^{鈭� 1} = (\begin{matrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{matrix})$

Step by step solution

Step 1:Definition for the B matrix of transformation T

Consider a linear transformation T from V to V, where V is an n-dimensional linear space .Let B be a basis of V.

Consider the linear transformation $L_{B} 鈭� T 鈭� {L_{B}}^{鈭� 1}$ from $R^{n}$ to $R^{n}$ with standard matrix B

Which implies $B \overset{鈫�}{x} = L_{B} [T ({L_{B}}^{鈭� 1} (\overset{鈫�}{x}))] 鈭赌 \overset{鈫�}{x} 鈥夆赌� i n 鈥夆赌� R^{n}$

This matrix B is called the B matrix of transformation T.

Step 2:Definition for the change of basis matrix of transformation T

Consider two bases U and B of an n-dimensional linear space V.

Consider the linear transformation $L_{U} 鈭� {L_{B}}^{鈭� 1}$ from $R^{n}$ to $R^{n}$ with standard matrix S,meaning that $S \overset{鈫�}{x} = L_{U} [{L_{B}}^{鈭� 1} (\overset{鈫�}{x})] 鈥夆赌� 鈭赌鈥夆赌夆€� \overset{鈫�}{x} 鈥夆赌� i n 鈥夆赌� R^{n}$

This invertible matrix S is called the change of basis matrix from B to U,sometimes denoted by $S_{B 鈫� U}$

(a) Solution for the change of matrix S

Consider the bases as B and U as follows

$\begin{array}{l} B = [\begin{array}{l} 1 鈥夆赌夆€夆赌夆€夆赌� 鈭� 1 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 1 \\ 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 2 鈥夆赌夆€夆赌夆€� 鈭� 4 \\ 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 4 \end{array}] \\ U = [\begin{array}{l} 1 鈥夆赌夆€夆赌夆€夆赌� 鈭� 1 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 1 \\ 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 2 鈥夆赌夆€夆赌夆€� 鈭� 4 \\ 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 4 \end{array}] \end{array}$

Now by inspection we find out the change of matrix from B to U

$\begin{array}{l} S_{B 鈫� U} = [\begin{array}{l} a 鈭� b + c \\ b 鈭� 2 c \\ c \end{array}] \\ S_{B 鈫� U} = (\begin{matrix} 1 & 鈭� 1 & 1 \\ 0 & 1 & 鈭� 2 \\ 0 & 0 & 1 \end{matrix}) \end{array}$

Hence the solution.

(b) Solution for the SB=AS

Consider the matrix as follows

$\begin{array}{l} B = [\begin{array}{l} 1 鈥夆赌夆€夆赌夆€夆赌� 鈭� 1 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 1 \\ 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 2 鈥夆赌夆€夆赌夆€� 鈭� 4 \\ 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 4 \end{array}] \\ A = [\begin{array}{l} 1 鈥夆赌夆€夆赌夆€夆赌� 鈭� 1 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 1 \\ 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 2 鈥夆赌夆€夆赌夆€� 鈭� 4 \\ 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 4 \end{array}] \end{array}$

Here both the A and B matrix is same.

$S = (\begin{matrix} 1 & 鈭� 1 & 1 \\ 0 & 1 & 鈭� 2 \\ 0 & 0 & 1 \end{matrix})$

$\begin{array}{l} S B = (\begin{matrix} 1 & 鈭� 1 & 1 \\ 0 & 2 & 鈭� 4 \\ 0 & 0 & 4 \end{matrix}) [\begin{array}{l} 1 鈥夆赌夆€夆赌夆€夆赌� 鈭� 1 鈥夆赌夆€夆赌夆€夆赌� 1 \\ 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 1 鈥夆赌夆€夆赌夆€夆赌� 鈭� 2 \\ 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 1 \end{array}] \\ S B = (\begin{matrix} 1 & 鈭� 2 & 5 \\ 0 & 2 & 鈭� 8 \\ 0 & 0 & 4 \end{matrix}) \end{array}$

Similarly,AS becomes as follows

$\begin{array}{l} A S = [\begin{array}{l} 1 鈥夆赌夆€夆赌夆€夆赌� 鈭� 1 鈥夆赌夆€夆赌夆€夆赌� 1 \\ 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 1 鈥夆赌夆€夆赌夆€夆赌� 鈭� 2 \\ 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 0 鈥夆赌夆€夆赌夆€夆赌夆€夆赌� 1 \end{array}] (\begin{matrix} 1 & 鈭� 1 & 1 \\ 0 & 2 & 鈭� 4 \\ 0 & 0 & 4 \end{matrix}) \\ A S = (\begin{matrix} 1 & 鈭� 2 & 5 \\ 0 & 2 & 鈭� 8 \\ 0 & 0 & 4 \end{matrix}) \end{array}$

Hence the proof

(c) Solution for the change of matrix from U to B

Consider the bases as B and U as follows

Now by inspection we find out the change of matrix from U to B

$\begin{array}{l} {S^{鈭� 1}}_{U 鈫� B} = [\begin{array}{l} a + b + c \\ b + 2 c \\ c \end{array}] \\ {S^{鈭� 1}}_{U 鈫� B} = (\begin{matrix} 1 & 1 & 1 \\ 0 & 1 & 2 \\ 0 & 0 & 1 \end{matrix}) \end{array}$

Hence the solution.

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Short Answer

Step by step solution

Step 1:Definition for the B matrix of transformation T

Step 2:Definition for the change of basis matrix of transformation T

(a) Solution for the change of matrix S

(b) Solution for the SB=AS

(c) Solution for the change of matrix from U to B

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91影视

a.Find the change of basis matrix S from the basis B considered in Exercise 28 to the standard basisU=(1,t,t2)ofP2considered in Exercise 27.b.Verify the formula SB=AS for the matrices B and A you found in Exercises 28 and 27,respectively.c.Find the change of basis matrix from U to B

Short Answer

Step by step solution

Step 1:Definition for the B matrix of transformation T

Step 2:Definition for the change of basis matrix of transformation T

(a) Solution for the change of matrix S

(b) Solution for the SB=AS

(c) Solution for the change of matrix from U to B

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Discrete Mathematics

Applied Mathematics

Decision Maths

Probability and Statistics

Mechanics Maths

Study anywhere. Anytime. Across all devices.