1 4 in the column space of -9 - 11 - 36 ? 3 2 12 O Yes O No 0 4 – 11 - 36 1 If "Yes," write as a linear combination of the columns of 9. | 3 2 12 1 4 36 %3D 3 2 12

Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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### Exercise on Column Spaces and Linear Combinations

**Question:**

Is the vector 

\[
\begin{bmatrix}
0 \\
-6 \\
1
\end{bmatrix}
\]

in the column space of the matrix 

\[
\begin{bmatrix}
1 & 0 & 4 \\
-9 & -11 & -36 \\
3 & 2 & 12
\end{bmatrix}
\]?

- ○ Yes
- ○ No

**If "Yes," write**

\[
\begin{bmatrix}
0 \\
-6 \\
1
\end{bmatrix}
\]

as a linear combination of the columns of 

\[
\begin{bmatrix}
1 & 0 & 4 \\
-9 & -11 & -36 \\
3 & 2 & 12
\end{bmatrix}
\].

The solution should be in the form:

\[
\begin{bmatrix}
0 \\
-6 \\
1
\end{bmatrix}
= 
\begin{bmatrix}
1 \\
-9 \\
3
\end{bmatrix}
\cdot \_\_ 
+
\begin{bmatrix}
0 \\
-11 \\
2
\end{bmatrix}
\cdot \_\_
+
\begin{bmatrix}
4 \\
-36 \\
12
\end{bmatrix}
\cdot \_\_
\]

Fill in the blanks with appropriate scalars to express the given vector as a linear combination of the columns of the matrix.
Transcribed Image Text:### Exercise on Column Spaces and Linear Combinations **Question:** Is the vector \[ \begin{bmatrix} 0 \\ -6 \\ 1 \end{bmatrix} \] in the column space of the matrix \[ \begin{bmatrix} 1 & 0 & 4 \\ -9 & -11 & -36 \\ 3 & 2 & 12 \end{bmatrix} \]? - ○ Yes - ○ No **If "Yes," write** \[ \begin{bmatrix} 0 \\ -6 \\ 1 \end{bmatrix} \] as a linear combination of the columns of \[ \begin{bmatrix} 1 & 0 & 4 \\ -9 & -11 & -36 \\ 3 & 2 & 12 \end{bmatrix} \]. The solution should be in the form: \[ \begin{bmatrix} 0 \\ -6 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ -9 \\ 3 \end{bmatrix} \cdot \_\_ + \begin{bmatrix} 0 \\ -11 \\ 2 \end{bmatrix} \cdot \_\_ + \begin{bmatrix} 4 \\ -36 \\ 12 \end{bmatrix} \cdot \_\_ \] Fill in the blanks with appropriate scalars to express the given vector as a linear combination of the columns of the matrix.
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