- #1

Bashyboy

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## Homework Statement

Find the column of the matrix in Exercise 39 that can be deleted and yet have the remaining matrix columns span [itex]\mathbb{R}^4[/itex]

## Homework Equations

The matrix from problem 39 is

[itex]\begin{bmatrix}

10 & -7 & 1 & 4 & 6 \\

-8 & 4 & -6 & -10 & -3 \\

-7 & 11 & -5 & -1 & -8 \\

3 & -1 & 10 & 12 & 12 \\

\end{bmatrix}

[/itex]

## The Attempt at a Solution

I had MATLAB compute the row-reduced echelon form of the matrix augmented with the arbitrary point/vector [itex]\mathbf{b}[/itex], where [itex]\mathbf{b} \in \mathbb{R}^4[/itex]. The result is given as an attachment.

I was wondering, does this augmented matrix contain any free variables. I believe it does, and allow me to explain:

The augmented matrix can be written as

[itex]\begin{bmatrix}

1 & 0 & 0 & 1 & 0 & f_1(b) \\

0 & 1 & 0 & 1 & 0 & f_2(b) \\

0 & 0 & 1 & 1 & 0 & f_3(b) \\

0 & 0 & 0 & 0 & 1 & f_4(b) \\

\end{bmatrix}[/itex]

Writing out the system of linear equations, that this augmented matrix represents, gives

[itex]\begin{array} \\

x_1 & + & x_2 & + & x_3 & + & x_4 & = & f_1(b) \\

& & x_2 & + & x_3 & & & = & f_2(b) \\

& & & & x_3 & + & x_4 & = & f_3(b) \\

& & & & & & x_4 & = & f_4(b) \\

\end{array}[/itex]

If I understand what a free variable is, it is one of which can be written in terms of the others. In this example, we have the other variables can be written as a function of the variables x_4. So, we the variable x_4 is free because you can choose any value of it, and find the corresponding b_i's will be; then from these b_i's, x_3, x_2, and x_1 can be determined.

Am I wrong?

#### Attachments

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