Proving Linear Independence of Elements in R^4

[JinM]

Homework Statement

Let v, x_1, x_2, x_3 be elements in R^4, and suppose that there are distinct real numbers c1, c2, and c3 such that v = c_1*x_1 + c_2*x_2 + c_2*x_2. Prove that x_1, x_2, and x_3 are independent.

The Attempt at a Solution

Let A=[x_1 x_2 x_3]. Then Col(A) = span{x_1, x_2, x_3}. By hypothesis, v belongs to Col(A), and v can be written as a linear combination of the x's with distinct constants that are not all zeros. **If at least one of the columns is a linear combination of the other columns, then v can be written as a linear combination of the other two x's with distinct coefficients. If one of the two coefficients is zero, then v = 0*x_2 + c_3*x_3 = 0*x_1 + 0*x_2 + c_3*x_3, which contradicts the hypothesis. Thus the X's are independent.

Does this actually prove the question? I'm not really familiar with proofs so I'm not sure of what I'm doing following the stars above. Thanks in advance! :)

 

[Defennder]

Did you state the question properly? Because it looks as though if x_1, x_2,x_3,v only have to satisfy \vec{v} = c_1\vec{x_1} + c_2\vec{x_2} + c_3\vec{x_3}, one can easily come up with counterexamples such that \vec{x_1}, \vec{x_2}, \vec{x_3} are linearly dependent. Was some condition omitted?

 

[HallsofIvy]

Homework Statement

Let v, x_1, x_2, x_3 be elements in R^4, and suppose that there are distinct real numbers c1, c2, and c3 such that v = c_1*x_1 + c_2*x_2 + c_2*x_2. Prove that x_1, x_2, and x_3 are independent.

As Defennder said, this is not true. As a counterexample, take x_1= <1, 1, 0, 0>, x_2= ,1, 0, 1, 0>, x_3= <2, 1, 1, 0> and v= <4, 2, 2, 0>. Then v= 1*x_1+ 1*x_2+ 1*x_3 but x_1, x_2, and x_3 are clearly not independent.

The Attempt at a Solution

Let A=[x_1 x_2 x_3]. Then Col(A) = span{x_1, x_2, x_3}. By hypothesis, v belongs to Col(A), and v can be written as a linear combination of the x's with distinct constants that are not all zeros. **If at least one of the columns is a linear combination of the other columns, then v can be written as a linear combination of the other two x's with distinct coefficients. If one of the two coefficients is zero, then v = 0*x_2 + c_3*x_3 = 0*x_1 + 0*x_2 + c_3*x_3, which contradicts the hypothesis. Thus the X's are independent.

Does this actually prove the question? I'm not really familiar with proofs so I'm not sure of what I'm doing following the stars above. Thanks in advance! :)