Solution Found!
Solved: In Exercises 1 and 2, you may assume that is an
Chapter 6, Problem 2E(choose chapter or problem)
\(\mathbf{u}_1=\left[\begin{array}{l} 1 \\ 2 \\ 1 \\ 1 \end{array}\right], \quad \mathbf{u}_2=\left[\begin{array}{r} -2 \\ 1 \\ -1 \\ 1 \end{array}\right], \quad \mathbf{u}_3=\left[\begin{array}{r} 1 \\ 1 \\ -2 \\ -1 \end{array}\right], \quad \mathbf{u}_4=\left[\begin{array}{r} -1 \\ 1 \\ 1 \\ -2 \end{array}\right]\)
\(\mathbf{v}=\left[\begin{array}{r}4 \\ 5 \\ -3 \\ 3\end{array}\right]\). Write \(\mathbf{v}\) as the sum of two vectors, one in \(\operatorname{Span}\left\{\mathbf{u}_1\right\}\) and the other in \(\operatorname{Span}\left\{\mathbf{u}_2, \mathbf{u}_3, \mathbf{u}_4\right\}\).
Questions & Answers
QUESTION:
\(\mathbf{u}_1=\left[\begin{array}{l} 1 \\ 2 \\ 1 \\ 1 \end{array}\right], \quad \mathbf{u}_2=\left[\begin{array}{r} -2 \\ 1 \\ -1 \\ 1 \end{array}\right], \quad \mathbf{u}_3=\left[\begin{array}{r} 1 \\ 1 \\ -2 \\ -1 \end{array}\right], \quad \mathbf{u}_4=\left[\begin{array}{r} -1 \\ 1 \\ 1 \\ -2 \end{array}\right]\)
\(\mathbf{v}=\left[\begin{array}{r}4 \\ 5 \\ -3 \\ 3\end{array}\right]\). Write \(\mathbf{v}\) as the sum of two vectors, one in \(\operatorname{Span}\left\{\mathbf{u}_1\right\}\) and the other in \(\operatorname{Span}\left\{\mathbf{u}_2, \mathbf{u}_3, \mathbf{u}_4\right\}\).
ANSWER:Solution 2EStep 1 of 3Write the vectors, and Suppose that and The objective is to write the vector as the sum of the element of and an element of .Let be in and .To show that , use orthogonal decomposition theorem.