Show that the columns of X are linearly independent if the constraint α1 = 0 is imposed.
时间: 2024-06-02 11:10:07 浏览: 16
To show that the columns of X are linearly independent if the constraint α1 = 0 is imposed, we need to show that the only solution to the equation Xα = 0 (where α is a vector of coefficients) is the trivial solution α = 0.
Let X be an n x p matrix with columns x1, x2, ..., xp. If we impose the constraint α1 = 0, then the equation Xα = 0 becomes:
0*x1 + α2*x2 + α3*x3 + ... + αp*xp = 0
This can be rewritten as:
α2*x2 + α3*x3 + ... + αp*xp = 0
Assume that the columns of X are linearly dependent, so there exist coefficients α2, α3, ..., αp, not all zero, such that α2*x2 + α3*x3 + ... + αp*xp = 0. Without loss of generality, we can assume that α2 ≠ 0.
Then we can solve for x2 in terms of x3, ..., xp as:
x2 = (-α3/α2)*x3 - ... - (αp/α2)*xp
Substituting this expression into the equation α2*x2 + α3*x3 + ... + αp*xp = 0, we get:
α2*(-α3/α2)*x3 - α2*(α4/α2)*x4 - ... - α2*(αp/α2)*xp + α3*x3 + ... + αp*xp = 0
Simplifying this expression, we get:
-α3*x3 - α4*x4 - ... - αp*xp + α3*x3 + ... + αp*xp = 0
This is a contradiction, since we assumed that not all of the coefficients α2, α3, ..., αp are zero, but we ended up with the equation 0 = 0. Therefore, our assumption that the columns of X are linearly dependent must be false, and we can conclude that the columns of X are linearly independent if the constraint α1 = 0 is imposed.
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