Linear Independence of Sets in a Linear Space

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.
  • #1
mrs.malfoy
3
0
Let V be a linear space and u, v, w [tex]\in[/tex] V. Show that if {u, v, w} is linearly independent then so is the set {u, u+v, u+v+w}
 
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  • #2
mrs.malfoy said:
Let V be a linear space and u, v, w [tex]\in[/tex] V. Show that U [tex]\cap[/tex] V is not equal to {OV}
?? What is U?
 
  • #3
Can you explain your notation?
So u is an element of the linear space V, but what is U?
What is OV, is it the zero element in V?
 

Related to Linear Independence of Sets in a Linear Space

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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