Linear Algebra Quiz
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VECTOR SPACES & SUBSPACES
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Let f1(x)=sin⁡(x)f_1(x) = \sin(x), f2(x)=cos⁡(x)f_2(x) = \cos(x), and f3(x)=sin⁡(x+π/4)f_3(x) = \sin(x + \pi/4) be functions in the vector space of continuous functions on R\mathbb{R}. A student wants to determine if {f1,f2,f3}\{f_1, f_2, f_3\} is linearly independent by checking if c1f1(x)+c2f2(x)+c3f3(x)=0c_1 f_1(x) + c_2 f_2(x) + c_3 f_3(x) = 0 has only the trivial solution. Using the identity sin⁡(x+π/4)=22(sin⁡(x)+cos⁡(x))\sin(x + \pi/4) = \frac{\sqrt{2}}{2}(\sin(x) + \cos(x)), what should the student conclude?
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