# Systematic way of extending a set to a basis

• negation
In summary: So is adding vectors that are not orthogonal. So is adding vectors that are even dependent on the original set. We're not looking for the shortest solution, just a solution.In summary, to extend the given set of vectors to R4, a fourth vector w is required that is linearly independent from u1, u2, and u3. One approach is to use trial-and-error, but another approach is to set up a system of equations by setting the fourth vector as (v1, v2, v3, v4) and solving for the values of v1 and v2 that make the fourth vector perpendicular to u1, u2, and u3. This may not always be the most efficient method, but it
negation

## Homework Statement

I want to extend the below U set of vectors to R4.
u1 = (0, 0, 0, -4), u2 = (0, 0, -4, 3), u3 = (3, 2, 3, -2).

## The Attempt at a Solution

For a set of vectors to form a basis for Rn, the vectors must be LI and spans Rn(has n vectors)

u1, u2 and u3 are linearly independent and spans R3 but not R4.
To extend u1, u2 and u3 to R4. I require another vector, w. $w\notin span{u1,u2,u3}$

One of the solution is trial-and-error which I am not keen.

I started with this:

0λ1 + 0λ2 +3λ3 +λ4w1= 0
0λ1 + 0λ2 +2λ3 +λ4w2= 0
0λ1 -4λ2 +3λ3 + λ4w3= 0
-4λ1 +3λ2 -2λ3 + λ4w4 = 0

which is a brick wall.

The other approach I had was UW = 0
I want to find the W set of vectors that maps the coefficient matrix U into the zero-vector.

In the second approach, how are you defining U and what's stopping you from doing it?

vela said:
In the second approach, how are you defining U and what's stopping you from doing it?

There's nothing stopping me from doing so. The calculation can easily be done but I'd appreciate if anyone could shed some light on the idea behind using the definition of UW=0 to solve for the W set of vectors.

negation said:
There's nothing stopping me from doing so. The calculation can easily be done but I'd appreciate if anyone could shed some light on the idea behind using the definition of UW=0 to solve for the W set of vectors.
Well, again, you need to define what U and W are first. We can't read your mind.

negation said:

## Homework Statement

I want to extend the below U set of vectors to R4.
u1 = (0, 0, 0, -4), u2 = (0, 0, -4, 3), u3 = (3, 2, 3, -2).

## The Attempt at a Solution

For a set of vectors to form a basis for Rn, the vectors must be LI and spans Rn(has n vectors)

u1, u2 and u3 are linearly independent and spans R3 but not R4.
To extend u1, u2 and u3 to R4. I require another vector, w. $w\notin span{u1,u2,u3}$

One of the solution is trial-and-error which I am not keen.

I started with this:

0λ1 + 0λ2 +3λ3 +λ4w1= 0
0λ1 + 0λ2 +2λ3 +λ4w2= 0
0λ1 -4λ2 +3λ3 + λ4w3= 0
-4λ1 +3λ2 -2λ3 + λ4w4 = 0

which is a brick wall.

The other approach I had was UW = 0
I want to find the W set of vectors that maps the coefficient matrix U into the zero-vector.

There is a way that is easy in this problem, but is not really systematic and applicable to all problems without modification. Say the fourth vector you want is ##v = (v_1,v_2,v_3,v_4)##. If you want v to be orthogonal to all the u1, u2 and u3, then ##v \perp u1 \rightarrow v_4 = 0##.
So, now ##v = (v_1,v_2,v_3,0)##. The condition ##v \perp u2 \rightarrow v_3=0##. So, now ##v = (v_1,v_2,0,0)##, and ##v \perp u3 \rightarrow 3v_1+2v_2 = 0##. Take any v_1 and v_2 that satisfy this last equation; then v is perpendicular to all the ui, so is linearly independent of them.

A basis need not be orthogonal. i.e. (1,1), (1,0) is a basis for R2

Mugged said:
A basis need not be orthogonal. i.e. (1,1), (1,0) is a basis for R2

Nobody said it was. But adding vectors orthogonal to the original set is an easy solution.

## What is a basis?

A basis is a set of linearly independent vectors that span a vector space. This means that every vector in the space can be written as a linear combination of the basis vectors.

## Why is it important to extend a set to a basis?

Extending a set to a basis is important because it allows us to fully describe the vector space and perform calculations on it. Also, a basis provides a more concise and efficient way to represent vectors in the space.

## What is a systematic way of extending a set to a basis?

The systematic way of extending a set to a basis is called the "stepping stone" method. It involves adding one vector at a time to the set and checking if the resulting set is linearly independent. If not, another vector is added and the process is repeated until a basis is obtained.

## Can any set be extended to a basis?

No, not all sets can be extended to a basis. The set must be linearly independent and span the vector space in order for it to be extended to a basis.

## Are there different ways to extend a set to a basis?

Yes, there are multiple ways to extend a set to a basis. These include the "stepping stone" method, the Gram-Schmidt process, and the pivot columns in a matrix. Each method has its own advantages and may be more suitable for certain types of vector spaces.

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