# 12.6 linearly dependent or linearly independent?

• MHB
• karush
In summary, the vectors v_1= x^2+ 1, v_2= x+ 2, and v_3= x^2+ 2x are linearly independent because the system of equations formed from their coefficients has only the trivial solution. The Wronskian method can also be used to determine linear independence for sets of functions.
karush
Gold Member
MHB
Are the vectors
$$v_1=x^2+1 ,\quad v_2=x+2 ,\quad v_3=x^2+2x$$
linearly dependent or linearly independent?
if
$$c_1(x^2+1)+c_2(x+2)+c_3(x^2+2x)=0$$
is the system
$$\begin{array}{rrrrr} &c_1 & &c_3 = &0\\ & &c_2 &2c_3= &0\\ &c_1 &2c_2& = &0 \end{array}$$
I presume at this point observation can be made that this linear dependent
but also...
$$\left[ \begin{array}{ccc|c} 1 & 0 & 1 & 0 \\ 0 & 1 & 2 & 0 \\ 1 & 2 & 0 & 0 \end{array} \right] \sim \left[ \begin{array}{ccc|c} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{array} \right]$$

Last edited:
With a set of functions, you normally use the Wronskian to determine linear independence.

"
I presume at this point observation can be made that this linear dependent"
Why would you presume that?
Perhaps it is just the way I learned systems of equations but I never want to change to matrices to solve systems of equations!

To determine whether or not $$v_1= x^2+ 1$$, $$v_2= x+ 2$$, and $$v_3= x^2+ 2x$$ are independent or dependent we need to decide if there exist numbers, a, b, and c, no all 0, such that $$av_1+ bv_2+ cv_3= a(x^2+ 1)+ b(x+ 2)+ c(x^2+ 2x)= (a+ c)x^2+ (b+ 2c)x+ (a+ 2b)= 0$$.

In order that a polynomial be 0 for all x, all coefficients must be 0 so we must have
a+ c= 0
b+ 2c= 0
a+ 2b= 0

From the first equation c= -a so the second equation can be written as b+ 2(-a)= b- 2a= 0. Then b= 2a so the third equation is a+ 2(2a)= 5a= 0. a= 0 so b= 2(0)= 0 and c= -0= 0. The only solution is a= b= c= 0 so the vectors are independent.

Ackbach said:
With a set of functions, you normally use the Wronskian to determine linear independence.

ok we haven't done that yet

## 1. What is the difference between linearly dependent and linearly independent?

Linearly dependent means that one vector in a set of vectors can be written as a linear combination of the others. Linearly independent means that no vector in the set can be written as a linear combination of the others.

## 2. How can I determine if a set of vectors is linearly dependent or linearly independent?

To determine if a set of vectors is linearly dependent or linearly independent, you can use the method of Gaussian elimination or the determinant test. If the determinant of the matrix formed by the vectors is equal to 0, then the vectors are linearly dependent. If the determinant is not equal to 0, then the vectors are linearly independent.

## 3. Can a set of 3 vectors be linearly dependent?

Yes, a set of 3 vectors can be linearly dependent. In fact, any set of n vectors in an n-dimensional space can be linearly dependent.

## 4. What is the significance of linearly dependent or linearly independent vectors?

Linearly dependent or linearly independent vectors are important in linear algebra and other areas of mathematics because they help us understand the relationships between vectors and their spans. They also have applications in fields such as physics, engineering, and computer science.

## 5. Can linearly independent vectors be linearly dependent in a different vector space?

Yes, linearly independent vectors in one vector space can be linearly dependent in a different vector space. This is because the concept of linear dependence or independence depends on the dimension and basis of the vector space.

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