Is the only way to check if a set of vectors are linearly indep. is if the det does not equal 0? I have a problem where v1= [1;2;1;-1] v2=[0;2;1;1] v3=[3 1 0 2] but this creates a 3X4 matrix and you can't find the det of this.

I am really confused! thanks :D

Determining if a set of vectors is linearly independent does not necessarily require calculating the determinant. In general, to check if a set of vectors is linearly independent, you need to examine the coefficients of the vectors when they are combined linearly and set equal to zero.

Let's consider your example of v1=[1;2;1;-1], v2=[0;2;1;1], and v3=[3;1;0;2]. To check if these vectors are linearly independent, we need to determine if there are any non-trivial solutions to the linear combination:

a * v1 + b * v2 + c * v3 = 0,

where a, b, and c are scalars not all equal to zero.

To solve this system of equations, we can write it in matrix form using the given vectors as columns:

[1 0 3] [a] [0]
[2 2 1] [b] = [0]
[1 1 0] [c] [0]
[-1 1 2] [0]

The augmented matrix above represents a homogeneous system of linear equations. We can row-reduce this matrix to determine if there are non-trivial solutions (other than a = b = c = 0).

Performing row reduction, we get the following reduced row-echelon form:

[1 0 3 | 0 ]
[0 1 1 | 0 ]
[0 0 -3 | 0 ]
[0 0 0 | 0 ]

From the reduced row-echelon form, we can see that there is a row of zeros, which implies that any values of a, b, and c will satisfy the system of equations.

Therefore, the vectors v1, v2, and v3 are linearly dependent, not linearly independent.

So, the conclusion is that the given vectors are linearly dependent, which means they can be expressed as linear combinations of each other.