QuestionIs vector $b$ a linear combination of $a_{1}, a_{2}, a_{3}$ ? Choose A, B, C, or D based on the echelon matrix pivots.

Studdy Solution

STEP 1

Assumptions1. The vectors $a_{1}, a_{}, a_{3}$ and $b$ are given as $a_{1}=\left[\begin{array}{r} 1 \\ -4 \\ 0\end{array}\right], a_{}=\left[\begin{array}{l} 0 \\ 1 \\ 3\end{array}\right], a_{3}=\left[\begin{array}{r} 4 \\ -4 \\ 36\end{array}\right], b=\left[\begin{array}{r} 3 \\ -1 \\ 33\end{array}\right]$ . We need to determine if $b$ is a linear combination of $a_{1}, a_{}$ , and $a_{3}$ .

STEP 2

To determine if $b$ is a linear combination of $a_{1}, a_{2}$ , and $a_{}$ , we need to solve the following system of linear equations for scalars $x_{1}, x_{2}, x_{}$ $x_{1}a_{1} + x_{2}a_{2} + x_{}a_{} = b$

STEP 3

We can write this system of linear equations in matrix form as $\begin{bmatrix}1 &0 & \\ - &1 & - \\0 &3 &36 \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \\ x_{3} \end{bmatrix} = \begin{bmatrix}3 \\ -1 \\33 \end{bmatrix}$

STEP 4

Now, we will augment the matrix with vector $b$ to form the augmented matrix $\left[\begin{array}{ccc|c}1 &0 &4 &3 \\ -4 &1 & -4 & -1 \\0 &3 &36 &33 \end{array}\right]$

STEP 5

We will now use Gaussian elimination to bring this matrix to row echelon form. First, we will swap the first and second rows to bring a non-zero entry to the top left of the matrix $\left[\begin{array}{ccc|c} -4 &1 & -4 & -1 \\1 &0 &4 &3 \\0 &3 &36 &33 \end{array}\right]$

STEP 6

Next, we will multiply the first row by $-1/4$ to make the first entry of the first row1 $\left[\begin{array}{ccc|c}1 & -1/4 &1 &1/4 \\1 &0 &4 &3 \\0 &3 &36 &33 \end{array}\right]$

STEP 7

Subtract the first row from the second row to make the first entry of the second row0 $\left[\begin{array}{ccc|c}1 & -1/4 &1 &1/4 \\0 &1/4 &3 &11/4 \\0 &3 &36 &33 \end{array}\right]$

STEP 8

Multiply the second row by4 to make the second entry of the second row1 $\left[\begin{array}{ccc|c}1 & -1 &4 &1 \\0 &1 &12 &11 \\0 &3 &36 &33 \end{array}\right]$

STEP 9

Subtract3 times the second row from the third row to make the second entry of the third row $\left[\begin{array}{ccc|c} & - &4 & \\ & &12 &11 \\ & & & \end{array}\right]$

STEP 10

Now the matrix is in row echelon form. We can see that the pivots (leading entries) are in the first entry of the first column, the second entry of the second column, and there is no pivot in the third column. This means that the system of equations has a solution, and therefore $b$ is a linear combination of $a_{}, a_{2}$ , and $a_{3}$ .
The correct answer is B. Vector $\mathbf{b}$ is a linear combination of $\mathbf{a}_{}, \mathbf{a}_{2}$ , and $\mathbf{a}_{3}$ . The pivots in the corresponding echelon matrix are in the first entry in the first column and the second entry in the second column.

Was this helpful?

Get Studdy

Scan QR code with your device to get Studdy on iOS or Android

QR code

Studdy solves anything!

Download the app

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Parents Influencer program Contact Policy Terms