Are These Vectors Linearly Independent?

[Lynne]

Homework Statement

Given vectors:
a1 = (1; 2; 0),
a2 = (2; 1; 3),
a3 = (0; 3; -3).
Find out if these vectors are linearly independent.

Homework Equations

The Attempt at a Solution

$$\begin{cases}<br /> \lambda_1+2\lambda_2=0;\\<br /> 2\lambda+\lambda_2+3\lambda_3=0;\\<br /> 2\lambda_2-3\lambda_3=0;\\<br /> \end{cases}\\<br /> <br /> \lambda_1=\lambda_2=\lambda_3=0$$

$$D=\begin{vmatrix} 1 & 2 & 0 \\ 2 & 1 & 3 \\ 0 & 3 & -3 \end{vmatrix}=0$$

Vectors are not linearly independent because determinant is zero.
Am I correct?

 

[Defennder]

Yes, looks ok. It's also possible to do it by inspection since there are so few vectors involved. It shouldn't be too hard to spot that 2a1-a2=a3.

 

[rock.freak667]

yes that is correct.

 

[NoMoreExams]

Yes, looks ok. It's also possible to do it by inspection since there are so few vectors involved. It shouldn't be too hard to spot that 2a1-a2=a3.

Definitely agree with this approach. Determinants are thrown in too early without students understanding what they mean. It's best to seek a solution without them at an early level. Also seek Axler's paper :)

 

[sutupidmath]

Homework Statement

Given vectors:
a1 = (1; 2; 0),
a2 = (2; 1; 3),
a3 = (0; 3; -3).
Find out if these vectors are linearly independent.

Homework Equations

The Attempt at a Solution

$$\begin{cases}<br /> \lambda_1+2\lambda_2=0;\\<br /> 2\lambda+\lambda_2+3\lambda_3=0;\\<br /> 2\lambda_2-3\lambda_3=0;\\<br /> \end{cases}\\<br /> <br /> \lambda_1=\lambda_2=\lambda_3=0$$

$$D=\begin{vmatrix} 1 & 2 & 0 \\ 2 & 1 & 3 \\ 0 & 3 & -3 \end{vmatrix}=0$$

Vectors are not linearly independent because determinant is zero.
Am I correct?

Well yeah, since the determinant is zero, it means that the corresponding matrix is singular, so the column vectors of that matrix are linearly dependent, in which case your vectors actually consist of the columns of the matrix.
i.e
A=[a1,a2,a3] where a1,a2,a3 are column vectors you were given.

Another way of doing it is taking the dependence relation

$$x_1a_1+x_2a_2+x_3a_3=\bar 0$$ and solving this vector equation, and observing that there are nontrivial solutions to this vector equation, which actually meanst that the three vectors given are lin. dependent.

 

[Lynne]

Thank you very much.