MATHEMATICS COLLEGE

Show that W is a subspace of R^3.
Show that W is a subspace of R^3. - 1

Answers

Answer 1
Answer:

Answer:

Check the two conditions of Subspace.

Step-by-step explanation:

If W is a Subspace of a vector space, V then it should satisft the following conditions.

1) The zero element should be in W.

Zero element can be different for different vector spaces. For examples, zero vector in $ \math{R^2} $ is (0, 0) whereas, zero element in $ \math{R^3} $ is (0, 0 ,0).

2) For any two vectors, $ w_1 $ and $ w_2 $ in W, $ w_1 + w_2 $ should also be in W.

That is, it should be closed under addition.

3) For any vector $ w_1 $ in W and for any scalar, $ k $ in V, $ kw_1 $ should be in W.

That is it should be closed in scalar multiplication.

The conditions are mathematically represented as follows:

1) 0$ \in $ W.

2) If $ w_1 \in W; w_2 \in W $ then $ w_1 + w_2 \in W $.

3) $ \forall k \in V, and \hspace{2mm} \forall w_1 \in W \implies kw_1 \in W

Here V = $ \math{R^3} $ and W = Set of all (x, y, z) such that $ x - 2y + 5z = 0 $

We check for the conditions one by one.

1) The zero vector belongs to the subspace, W. Because (0, 0, 0) satisfies the given equation.

i.e., 0 - 2(0) + 5(0) = 0

2) Let us assume $ w_1 = (x_1, y_1, z_1) $ and $ w_2 = (x_2, y_2, z_2) $ are in W.

That means: $ x_1 - 2y_1 + 5z_1 = 0 $ and

$ x_2 - 2y_2 + 5z_2 = 0 $

We should check if the vectors are closed under addition.

Adding the two vectors we get:

$ w_1 + w_2 = x_1 + x_2 - 2(y_1 + y_2) + 5(z_1 + z_2) $

$ = x_1 + x_2 - 2y_1 - 2y_2 + 5z_1 + 5z_2 $

Rearranging these terms we get:

$ x_1 - 2y_1 + 5z_1 + x_2 - 2y_2 + 5z_2 $

So, the equation becomes, 0 + 0 = 0

So, it s closed under addition.

3) Let k be any scalar in V. And $ w_1 = (x, y, z) \in W $

This means $ x - 2y + 5z = 0 $

$ kw_1 = kx - 2ky + 5kz $

Taking k common outside, we get:

$ kw_1 = k(x - 2y + 5z) = 0 $

The equation becomes k(0) = 0.

So, it is closed under scalar multiplication.

Hence, W is a subspace of $ \math{R^3} $.

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Answers

Answer:

D. SSS

Step-by-step explanation:

Was given to us that the corresponding sides are congruent so is SSS.

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It takes a book editor 28 minutes to edit 6 pages. Find how long it takes her to edit 45 pages

Answers

Answer:

Around 9.57 minutes to edit

Step-by-step explanation:

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Which expression has a negative value?−3×3×−3×−3×−3

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−3×3×−3×−3

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Answers

Step-by-step explanation:

A - 3×3×-3×-3×-3 = 243

B - 3×-3×3 = 27

C - 3×3×-3×-3 = -81

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Answer: answer is c

Step-by-step explanation:

What value of x makes the equation x^3=64 true?

Answers

Answer:

x=4

Step-by-step explanation:

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A sample of 31 boxes of cereal has a sample standard deviation of 0.81 ounces. Construct a 93% confidence interval to estimate the true standard deviation of the fllins process for the boxes of cereal. a (0433, 1.120) b)(0.658,1.058) c (0.525, 1095) d (0.731,0.731) e) None of the above.

Answers

Final answer:

Using the chi-square distribution and given data, a 93% confidence interval for the standard deviation is found to be (0.525, 1.095). Therefore, the correct answer is option (c).

Explanation:

To construct a confidence interval to estimate the true standard deviation of the filling process for the boxes of cereal, we will use the chi-square distribution as it is used when dealing with standard deviation and variance.

The formula for the confidence interval in this case is: (sqrt((n-1)*s^2/chi2(upper)), sqrt((n-1)*s^2/chi2(lower))), where n is the sample size, s is the sample standard deviation and chi2 is the chi-square value for the corresponding degrees of freedom and confidence level.

In this case, we have n=31, s=0.81. The degrees of freedom are one less than the sample size, so df = 31 - 1 = 30. For a 93% confidence level, the chi-square values are X2(0.035, 30) = 21.92 and X2(0.965, 30) = 41.34.

Substituting these values into the formula, we get: (sqrt((31-1)*(0.81)^2/41.34), sqrt((31-1)*(0.81)^2/21.92)) = (0.525, 1.095). Hence, the correct answer is (c) (0.525, 1.095).

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Final answer:

The 93% confidence interval for the standard deviation of the filling process for the boxes of cereal is (0.658,1.058), which is choice (b). The solution is determined using chi-square distribution and the formula for constructing a confidence interval specifically for standard deviations.

Explanation:

To answer this question, we will use chi-square distribution which is commonly used to construct a confidence interval for standard deviations.

The formula for a confidence interval in this case is ( √((n-1)s² / χ²_(1 - α/2)), √((n-1)s² / χ²_(α/2)) ). Here, n is the sample size, s is the sample standard deviation, α is the significance level (1 - confidence level). χ² represents chi-square values, where α/2 and 1 - α/2 are the confidence levels for a two-tailed test.

Using the chi-square table, for n - 1 = 30 degrees of freedom, we find the chi-square value for (1 - α/2)=0.935 is 17.708 and the chi-square value for α/2=0.065 is 46.194. Substituting n = 31, s = 0.81 into the formula, we get the 93% confidence interval for the standard deviation of the filling process for the boxes of cereal as ( √((30*0.81²) / 46.194), √((30*0.81²) / 17.708) ) = (0.658, 1.058).

So, the correct answer is (b) (0.658,1.058).

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Consider the initial value problem:y' + 5/3y =1 - 1/5t, y(0)= yo

What equation expresses the requirement that the solution touches the t-axis?
a. y(t)= 0
b. y'(t)= 0
c. y''(t)= 0

Answers

Answer:

a. y(t) = 0

Step-by-step explanation:

There are two axis on the graph. One is x-axis which is horizontal line on the graph and the other is y-axis which is vertical side of the graph. The point where x-axis and y-axis meet is origin which has value 0. The equation to write the points of the graph is represented by y(x) = 0. In the given equation there is t variable used in the values.

Final answer:

The requirement that the solution of the given initial value problem 'touches' the t-axis is represented by the equation y(t) = 0. This is because the output of the function is zero at that specific value of t. Contrastingly, y'(t) = 0 and y''(t) = 0 indicate conditions of slope and rate of slope change.

Explanation:

In the given initial value problem, the requirement that the solution 'touches' the t-axis is represented by the equation y(t) = 0. This is because when the function Touches the t-axis, the y-value (output of the function) is zero for that specific value of t.

It's worth noting that y'(t) = 0 and y''(t) = 0 represent the conditions where the slope of a function is zero (which corresponds to a localminimum or maximum), and where the rate of change of the slope is zero (which can indicate a point of inflection), respectively.

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