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Homework Statement
Let [itex]v \in \mathbb{R}^n[/itex], let [itex]\{u_1, \ldots, u_k\}[/itex] be an orthonormal subset of [itex]\mathbb{R}^n[/itex] and let [itex]c_i[/itex] be the coefficient of the projection of v to the span of [itex]u_i[/itex]. Show that [itex]\|v\|^2 \ge c_1^2 + \cdots + c_k^2[/itex].
The attempt at a solution
[itex]c_i = v \cdot u_i[/itex] and [itex]\| v \|^2 = v \cdot v[/itex] so I can write the inequality as
[tex]v \cdot (v - (u_1 + \cdots + u_k)) \ge 0[/tex]
This means the angle between v and [itex]v - (u_1 + \cdots + u_k)[/itex] is less than 90 degrees. This is all I've been able to conjure. I'm trying to reverse-engineer the inequality back to something I know is true. Is this a good approach? Is there a better approach?
Let [itex]v \in \mathbb{R}^n[/itex], let [itex]\{u_1, \ldots, u_k\}[/itex] be an orthonormal subset of [itex]\mathbb{R}^n[/itex] and let [itex]c_i[/itex] be the coefficient of the projection of v to the span of [itex]u_i[/itex]. Show that [itex]\|v\|^2 \ge c_1^2 + \cdots + c_k^2[/itex].
The attempt at a solution
[itex]c_i = v \cdot u_i[/itex] and [itex]\| v \|^2 = v \cdot v[/itex] so I can write the inequality as
[tex]v \cdot (v - (u_1 + \cdots + u_k)) \ge 0[/tex]
This means the angle between v and [itex]v - (u_1 + \cdots + u_k)[/itex] is less than 90 degrees. This is all I've been able to conjure. I'm trying to reverse-engineer the inequality back to something I know is true. Is this a good approach? Is there a better approach?