Write **x** as the sum of two vectors, one in Span ${\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3}$ and one in Span ${\mathbf{u}_4}$. Assume that ${\mathbf{u}_1,\ldots,\mathbf{u}_4}$ is an orthogonal basis for $\mathbb{R}^4$.\[\mathbf{u}_1 = \begin{pmatrix}0 \\1 \\-7 \\-1 \\\end{pmatrix},\quad\mathbf{u}_2 = \begin{pmatrix}6 \\8 \\1 \\1 \\\end{pmatrix},\quad\mathbf{u}_3 =\begin{pmatrix}1 \\0 \\1 \\-7 \\\end{pmatrix},\quad\mathbf{u}_4 = \begin{pmatrix}8 \\-6 \\-1 \\1 \\\end{pmatrix},\quad\mathbf{x} = \begin{pmatrix}11 \\-5 \\4 \\0 \\\end{pmatrix}\]

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Answered: Write x as the sum of two vectors, one…

Write x as the sum of two vectors, one in Span {u, ,u2.u3} and one in Span fu4}. Assume that {u, .u4} is an orthogonal basis for R*. 1 8 11 - 6 8 U2 = 1 -7 - 1 4 1 1

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Concept explainers

Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.

Question

Write **x** as the sum of two vectors, one in Span ${\mathbf{u}_1, \mathbf{u}_2, \mathbf{u}_3}$ and one in Span ${\mathbf{u}_4}$. Assume that ${\mathbf{u}_1,\ldots,\mathbf{u}_4}$ is an orthogonal basis for $\mathbb{R}^4$.

\[
\mathbf{u}_1 =
\begin{pmatrix}
0 \\
1 \\
-7 \\
-1 \\
\end{pmatrix},
\quad
\mathbf{u}_2 =
\begin{pmatrix}
6 \\
8 \\
1 \\
1 \\
\end{pmatrix},
\quad
\mathbf{u}_3 =
\begin{pmatrix}
1 \\
0 \\
1 \\
-7 \\
\end{pmatrix},
\quad
\mathbf{u}_4 =
\begin{pmatrix}
8 \\
-6 \\
-1 \\
1 \\
\end{pmatrix},
\quad
\mathbf{x} =
\begin{pmatrix}
11 \\
-5 \\
4 \\
0 \\
\end{pmatrix}
\]

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