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Chapter 3: Problem 11

Consider the vectors \(\cos (x+\alpha)\) and \(\sin x\) in \(C[-\pi, \pi] .\) For what values of \(\alpha\) will the two vectors be linearly dependent? Give a graphical interpretation of your answer.

Short Answer

Expert verified
The vectors $\cos(x+\alpha)$ and $\sin x$ are linearly dependent in $C[-\pi, \pi]$ when there exists a non-zero constant $k$ such that $k \cdot \cos(x + \alpha) = \sin(x)$. We can relate these functions using the identity $\sin(x + \pi/2) = \cos(x)$ and find that for $k=1$, we get $\cos(x + \alpha) = \sin((x + \alpha) + \pi/2)$. Solving for $\alpha$, we obtain that $\alpha$ can take any value in the range [-蟺/2, 3蟺/2]. The graphical interpretation of this result is that the graphs of $\cos(x + \alpha)$ and $\sin(x)$ will be the same for each value of $\alpha$ in the specified range, and the two functions are obtained from each other by a horizontal shift corresponding to the $\alpha$ value found. The shape, amplitude, and period of the functions remain unchanged.

Step by step solution

01

Define linear dependence

Two vectors are linearly dependent if one can be obtained from the other by multiplying with a constant k, which is not equal to zero. In this case, we are looking for a constant k and a value of 伪 for which: k * cos(x + 伪) = sin(x)
02

Analyze the given functions

We have two trigonometric functions: cos(x+伪) and sin(x). To make them linearly dependent, we need to somehow relate the sine and cosine functions. Recall the identity: sin(x + 蟺/2) = cos(x)
03

Relate the given functions using the identity

We can relate the given functions using the identity we mentioned in step 2. To achieve this, we need to replace x with (x + 伪) in the identity: sin((x + 伪) + 蟺/2) = cos(x + 伪) Now, we can see that if we choose k = 1, we get: cos(x + 伪) = sin((x + 伪) + 蟺/2 )
04

Find the values of 伪

From step 3, we can see that cos(x + 伪) and sin(x) are linearly dependent if and only if: sin(x) = sin((x + 伪) + 蟺/2) We can conclude that 伪 must satisfy the equation: (x + 伪) + 蟺/2 = x Solving for 伪, we get: 伪 = 蟺/2 - x The values of 伪 decided by x are in the range of [-蟺, 蟺]. Therefore, 伪 can take any value in the range of [-蟺/2, 3蟺/2].
05

Graphical interpretation

A graphical interpretation of the result can be given by plotting the function sin(x) and cos(x + 伪) on the XY-plane within the interval [-蟺, 蟺]. For each value of 伪 in the range [-蟺/2, 3蟺/2], the graphs of cos(x + 伪) and sin(x) will be the same since they are linearly dependent. The plot will show that two functions are obtained from each other by a horizontal shift corresponding to the 伪 value found. The shape of the functions, as well as the amplitude and period, will remain the same for all values of 伪 in the specified range.

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