- #1

Samuelb88

- 162

- 0

## Homework Statement

The set of vectors

**u**= {1,-2,2,1},

**v**= {1,3,1,1},

**w**= {3,4,4,3} cannot span R4. Complete this set to create a set of vectors that will span R4. Show that your set of vectors spans R4.

## The Attempt at a Solution

Let [tex]y = {y_1,y_2,y_3,y_4}[/tex]. I write span{u,v,w,y} as the coefficient matrix:

[1,1,3,

**y_1**

-2,3,4,

**y_2**

2,1,4,

**y_3**

1,1,3,

**y_4**]

Using the first row to produce zeros in each row below yields:

[1,1,3,

**y_1**

0,5,10,

**y_2**+2

**y_1**

0,-1,-2,

**y_3**-2

**y_1**

0,0,0,

**y_4**-

**y_1**]

Using the second row to produce zeros in the row below yields:

[1,1,3,

**y_1**

0,5,10,

**y_2**+2

**y_1**

0,0,0,5

**y_3**-9

**y_2**+2

**y_1**

0,0,0,

**y_4**-

**y_1**]

So to my understanding, it would seem given the set {u,v,w}, a fourth vector

**y**cannot be chosen so that the set {u,v,w,y} spans R4 since not every row can contain a pivot position in this case. Please correct me if I am wrong.

- Sam

PS sorry for the messy work - I don't know how to write matrices in latex.