# Verifying if Vector b is in Span of Vectors v1 and v2

• charmedbeauty
In summary, the homework statement is that vector b in the span of vectors v1,v2. However, the attempt at a solution gives a different result numerically. Specifically, R3=R3+R1 which seems wrong because that gives you 1 in the first column of R3, which you do not want.

## Homework Statement

is vector b in the span of vectors v1,v2? Give reasons.

## The Attempt at a Solution

v1=(1,4,0,-1)

v2=(2,7,-2,-3)

b= (-1,-1,7,4)

set up in matrix

(v1,v2| b)

and after row reduction I have

(1,0,0,0),(2,-1,0,0)|(-1,3,6,6)

matrix has no sltn

since 0x + 0y = 6

therefore b does not span v1 ,v2.

my row reduction steps are..

R4= R4+R1
R2=R2-4R1

R4=R4+R2
R3=R3+R1

...

is this right?

The general result (negative) is correct. However, I get a different result numerically. Specifically, R3 = R3 + R1 seems wrong because that gives you 1 n the first column of R3, which you do not want.

voko said:
The general result (negative) is correct. However, I get a different result numerically. Specifically, R3 = R3 + R1 seems wrong because that gives you 1 n the first column of R3, which you do not want.

ohh thanks I did not see that.

A more fundamental resolution is that a vector, b, is in the span of vectors v1 and v2 if and only if there exist scalars, a and b, such that b= av1+ bv2 (that's the definition of 'span').

That is, there must exist a and b such that a(1, 4, 0, -1)+ b(2, 7, -2, -3)= (a+ 2b, 4a+ 7b, -2b, -a- 3b)= (-1, -1, 7, 4) so that we must have
a+ 2b= -1
4a+7b= -1
-2b= 7
-a- 3b= 4.

Of course, what you are doing is using the 'augented' matrix to try to solve those equations. But in this simple case, you can recognize that the third equation, -2b= 7 gives b= -7/2 and so the fourth equation becoms -a+ 21/2= 4 so that a= 21/2- 4= (21- 8)/2= 13/2. Now put those values of x and y into the first and second equations to see if they satisfy them: a+ 2b= -1 becomes 13/2+ (-14/2)= -1/2, NOT -1 so we can stop.

## 1. How do you determine if a vector is in the span of two other vectors?

To determine if a vector b is in the span of vectors v1 and v2, you can perform the following steps:

• Set up a system of equations using the components of the vectors v1, v2, and b.
• Use Gaussian elimination to put the system of equations into reduced row echelon form.
• If the system has a solution, then vector b is in the span of v1 and v2.
• If the system has no solution, then vector b is not in the span of v1 and v2.

## 2. What does it mean for a vector to be in the span of two other vectors?

A vector b is in the span of vectors v1 and v2 if it can be written as a linear combination of v1 and v2. This means that there exist scalars c1 and c2 such that b = c1v1 + c2v2.

## 3. Can a vector be in the span of two linearly dependent vectors?

Yes, a vector can be in the span of two linearly dependent vectors. This is because linear dependence means that one vector can be written as a multiple of the other, so any vector in their span can also be written as a linear combination of the two vectors.

## 4. How does the dimension of the span of two vectors relate to the vectors themselves?

The dimension of the span of two vectors v1 and v2 is either 1 or 2. If the two vectors are linearly independent, then the span has dimension 2 and any vector in the span is uniquely determined by its components. If the two vectors are linearly dependent, then the span has dimension 1 and any vector in the span can be written as a multiple of the single vector.

## 5. Is verifying if a vector is in the span of two other vectors the same as checking if it is in the subspace spanned by those vectors?

Yes, verifying if a vector b is in the span of vectors v1 and v2 is equivalent to checking if b is in the subspace spanned by v1 and v2. This is because the span of two vectors is the set of all possible linear combinations of those two vectors, which forms a subspace.