# Dimension of the linear span of a vector

• lolimcool
In summary, the dimension of the linear span of the given vectors is 2. This is determined by performing row reduction on the matrix formed by the vectors and observing that the resulting matrix has 2 pivot columns, indicating that the vectors are independent and form a basis for their span.
lolimcool

## Homework Statement

find the dimension of the linear span of the given vectors
v1 = ( 2, -3, 1) v2 = ( 5, -8, 3) v3 = (-5, 9, -4)

## The Attempt at a Solution

so all i did was make it a matrix and put it in rref and i got
[1 0 0]
[0 1 0]
[0 0 1]

does this mean dimensions of the linear span = 3?

Yes, it does. What you have shown is that those three vectors are independent and independent vectors form a basis for their span.

HallsofIvy said:
Yes, it does. What you have shown is that those three vectors are independent and independent vectors form a basis for their span.

It would mean that. Except I think you did the row reduction wrong. Those vectors aren't linearly independent.

Dick said:
It would mean that. Except I think you did the row reduction wrong. Those vectors aren't linearly independent.

oops i did i did it again and got

1 0 5
0 1 -3
0 0 0

so dimension = 2?

lolimcool said:
oops i did i did it again and got

1 0 5
0 1 -3
0 0 0

so dimension = 2?

Yes, dimension=2.

## 1. What is the dimension of the linear span of a vector?

The dimension of the linear span of a vector is the minimum number of vectors needed to span the entire vector space. It represents the number of linearly independent vectors that make up the span.

## 2. How is the dimension of the linear span of a vector calculated?

The dimension of the linear span of a vector can be calculated by finding the rank of the matrix formed by the vectors, or by counting the number of non-zero rows in the reduced row echelon form of the matrix.

## 3. Can the dimension of the linear span of a vector be greater than the number of vectors in the set?

Yes, the dimension of the linear span of a vector can be greater than the number of vectors in the set. This is because the dimension also takes into account linear combinations of the vectors, not just the individual vectors themselves.

## 4. How does the dimension of the linear span of a vector relate to the concept of linear independence?

The dimension of the linear span of a vector is closely related to linear independence. If the dimension is equal to the number of vectors in the set, then the vectors are linearly independent. However, if the dimension is less than the number of vectors, then the vectors are linearly dependent.

## 5. Why is understanding the dimension of the linear span of a vector important in linear algebra?

Understanding the dimension of the linear span of a vector is important in linear algebra because it helps determine the size and structure of a vector space. It also allows for the identification of linearly independent vectors, which are crucial in solving systems of linear equations and other applications in mathematics and science.

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