Determinant Test: Linear Independence Of Vectors In R^N

{v1…vp} in R^n is linearly independent if and only if x1v1 … + xpvp = 0

has only the trivial solution (x1=0, xp=0)Ax=0 has only the trivial solution x=0Ax=0 has no free variablesA has a pivot in every row

in R^n implies that x1 = x2 = … = xp = 0.

Explanation:

To prove that {v1…vp} in R^n is linearly independent, we need to show that the only way to express 0 as a linear combination of these vectors is by setting all the coefficients to zero. That is, if we have:

x1v1 + x2v2 + … + xpvp = 0

then we must show that the only solution to this system is x1 = x2 = … = xp = 0.

To do this, we can use the fact that the vector equation above represents a system of linear equations in the coefficients x1, x2, …, xp. Specifically, we can write this system as:

x1v1 + x2v2 + … + xpvp = 0,

or, equivalently,

x1v1 = -x2v2 – … – xpvp.

Since {v1…vp} is a set of vectors, we know that each vector can be expressed as a column vector in R^n. Letting V be the matrix whose columns are v1, v2, …, vp, we can rewrite the above equation as:

$V\begin{bmatrix}
x_1\\
x_2\\
\vdots\\
x_p
\end{bmatrix}=\begin{bmatrix}
0\\
0\\
\vdots\\
0
\end{bmatrix}$.

Since we want to show that the only solution to this system is x1 = x2 = … = xp = 0, we need to find the determinant of the matrix V. If the determinant is nonzero, then the system has a unique solution, and that solution is x1 = x2 = … = xp = 0. If the determinant is zero, then the system has either zero or infinitely many solutions, which means that {v1…vp} is linearly dependent.

Therefore, we have:

{v1…vp} in R^n is linearly independent if and only if the determinant of the matrix whose columns are v1, v2, …, vp is nonzero.

Alternatively, we can use the fact that the determinant of the matrix V is equal to the volume of the parallelepiped spanned by the vectors {v1…vp}. If this volume is zero, then the vectors are linearly dependent and lie in a lower-dimensional subspace of R^n. If the volume is nonzero, then the vectors are linearly independent and span the entire space R^n.

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