# Linear Independence of Sets in a Linear Space

• mrs.malfoy
In summary, linear independence refers to the property of a set of vectors where no vector can be expressed as a linear combination of the others. This is determined by using the linear combination method, and is important in understanding relationships between vectors and in various fields of mathematics and science. A linearly independent set must contain two or more vectors, whereas a single vector is automatically linearly independent. The difference between linear independence and linear dependence is that a set is linearly independent if no vector can be expressed as a linear combination of the others, while a set is linearly dependent if at least one vector can be expressed as a linear combination of the others.
mrs.malfoy
Let V be a linear space and u, v, w $$\in$$ V. Show that if {u, v, w} is linearly independent then so is the set {u, u+v, u+v+w}

Last edited:
mrs.malfoy said:
Let V be a linear space and u, v, w $$\in$$ V. Show that U $$\cap$$ V is not equal to {OV}
?? What is U?

So u is an element of the linear space V, but what is U?
What is OV, is it the zero element in V?

## What is linear independence?

Linear independence refers to the property of a set of vectors in a linear space, where no vector can be expressed as a linear combination of the other vectors in the set. In other words, each vector in the set contributes a unique direction to the overall space.

## How do you determine if a set of vectors is linearly independent?

To determine if a set of vectors is linearly independent, you can use the linear combination method. This involves setting up a system of equations where each vector is multiplied by a coefficient, and then solving for those coefficients. If the only solution is when all coefficients are equal to 0, then the set is linearly independent.

## Why is linear independence important?

Linear independence is important because it allows us to understand and describe the relationships between vectors in a linear space. This concept is fundamental to many areas of mathematics and science, including linear algebra, physics, and engineering.

## Can a linearly independent set contain only one vector?

No, a linearly independent set must contain two or more vectors. This is because a single vector cannot be expressed as a linear combination of itself, making it automatically linearly independent.

## What is the difference between linear independence and linear dependence?

The difference between linear independence and linear dependence is that a set of vectors is linearly independent if no vector can be expressed as a linear combination of the other vectors, while a set is linearly dependent if at least one vector can be expressed as a linear combination of the others.

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