Suppose Smokey the Bear is camping on the Olympic peninsula and a fly gets trapped in his tent. Let the tent be represented by the polyhedron P with vertices (0, 0, 0), (0, 1, 0), (1, 1, 0), (1, 0, 0), (0, 1/2, 1),(1, 1/2, 1) and let the position of the fly be represented by the random vector (X, Y, Z). Assume that the position of the fly is uniformly randomly distributed within P. 1. Compute the marginal p.d.f. of X, Y, and Z individually 2. Are X, Y, Z independent? 3. Are X and Y independent? Are X and Z independent? Are Y and Z independent?

Q: The vector projection of M = 2ax + 3ay + 2az onto P = 3ax + 2ay + 3az is

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Q: Find u and usuch that the null space for the matrix U= 3 v= 0 -2 1 1 -3 -15 1 - 9 1 is span{u, }.

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Q: Cl,9)(2,7), (3,3), (42) the least sa uares straght 'ine

A: Given: 1,9,2,7,3,3,4,2

Q: 2. Determine whether the vectors v, are orthonormal. If not, normalize them to produce an…

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Q: Let X be an n-dimensional random vector and the random vector Y be defined as Y = AX + b, where A…

A: It is widely used in statistics . It has applications in almost all multivariate problems .

Q: Find the least-squares line y = Bo + B1x that best fits the points (-3, -4), (-1, 1), (1,5), (3, 8)

A: Use least square method to solve it

Q: 4. (5pts) Determine if the columns of 3 -2 5 -6 -1 A -6 0 3 4 7 span R'.

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Q: Find the projection matrix P, and the projection of b= (1,2,3) onto 2. Let A= -1 -1 the column space…

A: Given matrix is A=1001-1-1

Q: 19. Can each vector in R4 be written as a linear combination of the columns of the matrix A above?…

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Q: Ar+By+C Derive the normal equations for finding the least-squares plane z for the N points (x1, Y1,…

A: We are given a set of N points x1, y1, z1,...,xN, yN, zN and we have to derive the normal equations…

Q: Explain why v = ( 1, 0, -1) cannot be a row of A and also in the nullspace.

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Q: 8.) Find a vectorx such that 5x-2v=2(u-5x) SOLUTION

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Q: Pick two random vectors in R³ (no zeros - few repeated numbers) and call them u and 7. Compute the…

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Q: (a) Show that the vectors v1 = (1, 2, 3, 4), v2 = (0, 1, 0, – 1), and v3 = (1, 3, 3, 3), form a…

A: (a) The given vectors are: v1=(1,2,3,4), v2=(0,1,0,-1), v3=(1,3,3,3)

Q: Find the matrix P that multiplies | y| to give P = א א

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Q: 4. Derive the normal equations for finding the least-squares plane z = Ax+By+C for the N points (x1,…

A: The assumption is that the z component of the data is functionally dependent on the x and y…

Q: Q6) Find the matrix A such that the given set is Col(A) 2s+3t r+s-2t :r,s,treal 4r+s 3r-s-t

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Q: -5

A: From graph we see that the graph intersect x axis at the points x=-3 , x=-1, x=2 and the graph…

Q: How many of the vectors (n-tuples) (1 2 2 -1 ), (-4 0 0 -4), (-1 1 -2 -2 ), (-2 0 0 -2 ), (-3 -2 -2…

A: If for given vectors the zero vector can only be obtained by their trivial linear combination then…

Q: Find the projection projuv, where u = (1,3,-2) and v = (2,-1,3).

A: Since we kow that the projection of v on u is given by                projuv=u.v||u||2u

Q: Apply Gram-Schmidt to (1, 0, −1, 1), (2, 1, −1, 0) , (2, −1, −1, 3) to obtain an orthonormal set…

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Q: 7. Givep that D- then N(D) contains B . All vectors that All vectors that belong to both N(A) and…

A: Given: D=AB Find the vectors that ND contains. Definition (Null space): Let T be a matrix, then the…

Q: 3= b, ,b2} and C= (c,.c2} be bases for a vector space V, and suppose b, = - 6c, + 5c2 and b2 = - 4c,…

A: Two bases B and C of a vector space V are given. We need to find change of co-ordinate Matrix from B…

Q: 2. Find a matrix that projects onto the line / 2x - 5y = 0 and find the projection of 6= (1, -2)…

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Linear Algebra

In mathematics, problems can be solved by the arithmetic operators like addition, subtraction, multiplication, and division. In algebra, we represent the numbers by any letter called a variable. If these variables are of degree 1, then it is studied under linear algebra. The first-degree equations involving algebraic expressions are called linear equations. That is, if you represent it by drawing a graph you will get a straight line.