# For a set of vectors to span R4 question

• Samuelb88
In summary, The set of vectors u = {1,-2,2,1}, v = {1,3,1,1}, w = {3,4,4,3} cannot span R4. To create a set of vectors that will span R4, one must add at least two more vectors to the set. The Gram-Schmidt process can be used to construct a basis for R4, or one can choose two vectors from the standard basis for R4 to add to the set.
Samuelb88

## 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 $$y = {y_1,y_2,y_3,y_4}$$. 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+2y_1
0,-1,-2,y_3-2y_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+2y_1
0,0,0,5y_3-9y_2+2y_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.

The problem is that u, v, and w are linearly dependent. Have you learned about the Gram-Schmidt process for constructing a basis yet?

Mark44 said:
The problem is that u, v, and w are linearly dependent. Have you learned about the Gram-Schmidt process for constructing a basis yet?

No I haven't. To my understanding though, the Gram-Schmidt process creates a set (for this case) of vectors {u',v',w'} so that v' is orthogonal to u', w' is orthogonal to v' and I suppose u', v', and w' are linearly independent so I may choose a fourth vector y' to "complete" this set so that this completed set spans R4?

You can use G-S to construct all four vectors of a basis for R4.

Samuelb88 said:
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.
You're right. You can't add just one more vector and span R4. You need to add two or more because, as Mark noted, the three you have are linearly dependent.
PS sorry for the messy work - I don't know how to write matrices in latex.
You can use the bmatrix environment to produce matrices in LaTeX. Click on the one below to see what the code is.

$$\begin{bmatrix} 1 & 1 & 3 & y_1 \\ -2 & 3 & 4 & y_2 \\ 2 & 1 & 4 & y_3 \\ 1 & 1 & 3 & y_4 \end{bmatrix}$$

Don't include 0. Your u, v, and w are linearly dependent, which can be seen from the first 3 columns of your last matrix (but taken one step further).

$$\begin{bmatrix}1 & 0 & 1 \\0 & 1 & 2 \\0 & 0 & 0 \\0 & 0 & 0\end{bmatrix}$$

This says c1 + c3 = 0, and that c2 + 2c3= 0, where c1, cw, and c3 are the coefficients of u, v, and w. If you let c3 = 1, then c1 = -1, c2 = -2, and c3 = 1.

In other words -u -2v + w = 0, so it can be seen that anyone of these vectors can be solved for in terms of the other two.

Without knowing G-S, I suppose the best way is to pick two of the three vectors, say u and v, and then try to find two more vectors out of the set of standard basis vectors for R4 - {<1, 0, 0, 0>, <0, 1, 0, 0>, <0, 0, 1, 0>, <0, 0, 0, 1>} - so that you have four linearly independent vectors.

For your spanning set you can throw in w.

## What does it mean for a set of vectors to span R4?

For a set of vectors to span R4 means that every vector in the 4-dimensional space can be written as a linear combination of the given set of vectors. In other words, the set of vectors can reach every point in R4.

## How can I determine if a set of vectors spans R4?

To determine if a set of vectors spans R4, you can use the row reduction method or the Gauss-Jordan elimination method to convert the vectors into a matrix. If the matrix has a pivot in every row, then the set of vectors spans R4. Otherwise, the set does not span R4.

## Can a set of two vectors span R4?

No, a set of two vectors cannot span R4 as R4 is a 4-dimensional space and at least 4 linearly independent vectors are needed to span it. Any set with less than 4 vectors will span a subspace of R4.

## Can a set of four vectors not span R4?

Yes, a set of four vectors can fail to span R4 if the vectors are linearly dependent. This means that at least one of the vectors can be written as a linear combination of the other three. In this case, the set of vectors will only span a subspace of R4.

## How can I find a set of vectors that spans R4?

To find a set of vectors that spans R4, you can use the basis of R4 which is {(1,0,0,0), (0,1,0,0), (0,0,1,0), (0,0,0,1)}. Any linear combination of these basis vectors will span R4. Alternatively, you can use the row reduction method to find a set of linearly independent vectors from a larger set that spans R4.

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