qsn 5 e j and m Determine it the following is a subspace of the vector space of

Question

qsn 5 e j and m

Determine it the following is a subspace of the vector space of real-valued functions: a. F = { f | f is continuous on [a, b] } b. F = { f | f is differentiable on [a, b] } c. F = { f | f is integrable on [a, b] } d. F = { f | f(10) - 0 } e. F = { f | f is a linear combination of x and sin (x) } f. F = { f | f is a solution to: f'(x) + 5 f(x) = 0 } g. i. F = { f | f(0) - 2 } h. F = { f | f = a + b 2^x } i. F = { f | f is a polynomial of degree at most 2 } j. F = { f | f is a solution to: f"(x) + 4 f(x) = 0 } k. F = { f | f is a linear combination of sin(x) and cos(x) } l. F = f | the integral of: f(x) on { 0, 1 } is 0 } m. F = { f | the integral of: f(x) exp(x) on [a, b] is 0 }

Explanation / Answer

E) .. yes it is a subspace as 0 function can be written as linear combination of x and sinx

Also if f and g are two functions which are linear combination of x and sinx then f+g is also so .

Hence a subspace.

J) yes a subspace as 0 is a solution to it. Also linear combination of solutions of homogeeous differential equation is also a solution . And hemce a subspace.

M) yes a subspace as 0 function belongs to it. Also if f and g hold this property then definitely f+g also hold this property due to the linear property of integration

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