Math Answers

5. Let φ : V → W be a linear transformation of vector spaces over the field F. The

kernel of φ is by definition the set ker(φ) ⊂ V of vectors v in V such that φ(v) = 0.

The image of φ is the subset im(φ) of vectors w ∈ W for which there exists some

v ∈ V such that φ(v) = w.

(a) Show that the kernel of is a subspace of V .

(b) Show that the image of is a subspace of W.

(c) Show that is injective if and only if the kernel is 0.

Let {u1, u2, ..., un} be an orthogonal basis for a subspace W of R

n

, and let T : R

n → R

n

be defined by T(x) = projW (x). Show that T is a linear transformation.

Show that T(x1, x2, x3, x4) = 3x1 −7x2 + 5x4 is a linear transformation by finding the

matrix for the transformation. Then find a basis for the null space of the transforma￾tion.

A group of students got the following scores in a test:9,12,15,18,21 and 24. Consider samples of size 3thag can be drawn from this population. List all the possible samples and the corresponding determine and list all possible samples and the corresponding sample means

A group of students got the following scores in a test:9,12,15,18,21 and 24. Consider samples of size 3thag can be drawn from this population. List all the possible samples and the corresponding mean. Determine and list all possible samples and the corresponding sample means.

A group of students got the following scores in a test:6,9,12,15,18 and 21. Consider samples of size 3thag can be drawn from this population. List all the possible samples and the corresponding mean.

A lottery that pays off P300 000 000 is made available for 10 000 000 tickets. Each ticket costs P50. Suppose the variable Z gives the net winnings from playing the lottery. What is the expected gain for joining the lottery with only one ticket?

An analysis of the monthly allowances received by a sample of five engineers from a telco company has been conducted recently. The mean and median of the allowances is RM7000. A unimodal among the observations is RM12,000. Find the standard deviation if the allowances paid to each engineer were in full thousands

A population consists of six scores: 11,13,15,16,17 and 19. If a sampling distribution of size 3 is drawn from the population. Compute for the population of mean, variance,variance of the sampling distributions of the sample mean and standard error of the mean.

A coin tossed and a die is rolled. The outcome of the coin is recorded 1 when it show a head and 0 when it shows a tail. The random variable R gives the sum of the outcomes of the coin and the die. Compute the average value of the random variables.