Q34E Find an orthonormal basis of the... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q34E (page 224)

Find an orthonormal basis of the kernel of the matrix $A = [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \end{matrix}]$ .

Short Answer

Expert verified

The solution is $\{\overset{鈫�}{v_{1}} [\begin{matrix} 1 \\ 2 \\ 1 \\ 0 \end{matrix}], \overset{鈫�}{v_{2}} = [\begin{matrix} 2 \\ - 3 \\ 0 \\ 1 \end{matrix}]\}$ .

Step by step solution

`Explanation of the solution

Consider the matrix A as follows:

$A = [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \end{matrix}]$

Let鈥檚 find out the kernel of the matrix A.

$K = \{[\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] 鈭� R^{4} : [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \end{matrix}] [\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] = [\begin{matrix} 0 \\ 0 \end{matrix}]\}$

Now apply for solve the above linear system as follows:

$[\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \end{matrix}] \overset{R_{2} 鈫� R_{3} - R_{1}}{鈫�} [\begin{matrix} 1 & 1 & 1 & 1 \\ 0 & 2 & 3 & 4 \end{matrix}] \overset{R_{2} 鈫� R_{3} - R_{1}}{鈫�} [\begin{matrix} 1 & 0 & - 1 & - 2 \\ 0 & 1 & 2 & 3 \end{matrix}]$

So, the kernel is as follows:

$K = \{[\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] 鈭� R^{4} : x_{1} - x_{2} - 2 x_{4} = 0, x_{2} + 2 x_{3} + 3 x_{4} = 0\} K = \{[\begin{matrix} x_{1} \\ x_{2} \\ x_{3} \\ x_{4} \end{matrix}] 鈭� R^{4} : x_{1} - x_{2} - 2 x_{4}, x_{2} = - 2 x_{3} - 3 x_{4}\} K = \{[\begin{matrix} x_{3} + 2 x_{4} \\ - 2 x_{3} - 3 x_{4} \\ x_{3} \\ x_{4} \end{matrix}] 鈭� R^{4} : x_{3} + x_{4} 鈭� R\} K = \{x_{3} [\begin{matrix} 1 \\ - 2 \\ 1 \\ 0 \end{matrix}] + [\begin{matrix} 1 \\ - 2 \\ 1 \\ 0 \end{matrix}] : x_{3}, x_{4} 鈭� R\} K = span \{[\begin{matrix} 1 \\ - 2 \\ 1 \\ 0 \end{matrix}], [\begin{matrix} 2 \\ - 3 \\ 0 \\ 1 \end{matrix}]\}$

Thus, the basis for K is $\{{\overset{鈫�}{v}}_{1} = [\begin{matrix} 1 \\ - 2 \\ 1 \\ 0 \end{matrix}], {\overset{鈫�}{v}}_{2} = [\begin{matrix} 2 \\ - 3 \\ 0 \\ 1 \end{matrix}]\}$ .

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91影视

Find an orthonormal basis of the kernel of the matrix $A = [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \end{matrix}]$ .

Short Answer

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`Explanation of the solution

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91影视

Find an orthonormal basis of the kernel of the matrix A=[11111234].

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Find an orthonormal basis of the kernel of the matrix $A = [\begin{matrix} 1 & 1 & 1 & 1 \\ 1 & 2 & 3 & 4 \end{matrix}]$ .