problem 12 Consider the vectors cos x alpha and sin x in C

problem 12

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. Given the functions 2x and |x|, show that these two vectors are linearly independent in C [-1, 1]. the vectors are linearly dependent in C[0, 1]. Prove that any finite set of vectors that contains the zero vector must be linearly dependent. Let v_1 and v_2 be two vectors in a vector space V. Show that v_1 and v_2 are linearly dependent if and only if one of the vectors is a scalar multiple of the other. Prove that any nonempty subset of a linearly independent set of vectors {v_1, ..., v_n} is also linearly independent. Let A be an m times n matrix. Show that if A has linearly independent column vectors, then N (A) = {0}.

Solution

In C[-1,1]

Value of 2x is <-2,2>

Value of |x| is <1,1>

To show functions linearly independent determinant must be non-zero

|-2 1|

|2 1 |=-2-2=-4

2x and |x| are independent

b)

In C[0,1]

Value of 2x is <0,2>

Value of |x| is <0,1>

To show functions linearly dependent determinant must be zero

|0 0|

|2 1|=0-0=0

2x and |x| are dependent