Question

Let A = [1 5 ; 3 1 ; 2 -4] and b = [1 ; 0 ; 3] (where semicolons represent a new row)

Is equation Ax=b consistent?

Let b(hat) be the orthogonal projection of b onto Col(A). Find b(hat).

Let x(hat) the least square solution of Ax=b. Use the formula x(hat) = (A^(T)A)^(−1) A^(T)b to compute x(hat). (A^(T) is A transpose)

Verify that x(hat) is the solution of Ax=b(hat).

Answer #1

1) Let a = 〈4,−5,−2〉 and b = 〈2,−4,−5〉 Find the projection of b
onto a. proj a b=
2) Find the area of a triangle PQR, where P=(−2,−4,0),
Q=(1,2,−1), and R=(−3,−6,5)
3) Complete the parametric equations of the line through the
points (7,6,-1) and (-4,4,8)
x(t)=7−11
y(t)=
z(t)=

Let M = ( (−3 1 3 4), (1 2 −1 −2), (−3 8 4 2))
14. (3 points) Let B1 be the basis for M you found by row
reducing M and let B2 be the basis for M you found by row reducing
M Transpose . Find the change of coordinate matrix from B2 to
B1.

T: R^3 ----> R^5 such that T(x), then...
a. A^-1 is 3x 5 matrix
b. the mapping cannot be onto
c. the set of solutions to Ax=0 is infinite
d. A has at least 2 free variables
e. none of the above

Consider W = Span{p1(t),p2(t)} where p1(t) = 1−t^2 and p2(t) =
3+2t are the polynomials deﬁned on the interval [−1,1]. Find the
orthogonal projection of q(t) = t^2−t−1 onto W.

Let V be the vector space of 2 × 2 matrices over R, let <A,
B>= tr(ABT ) be an inner product on V , and let U ⊆ V
be the subspace of symmetric 2 × 2 matrices. Compute the orthogonal
projection of the matrix A = (1 2
3 4)
on U, and compute the minimal distance between A and an element
of U.
Hint: Use the basis 1 0 0 0
0 0 0 1
0 1...

Consider the following.
u =
−6, −4, −7
, v =
3, 5, 2
(a) Find the projection of u onto
v.
(b) Find the vector component of u orthogonal to
v.

U= [2,-5,-1] V=[3,2,-3] Find the orthogonal projection of u onto
v. Then write u as the sum of two orthogonal vectors, one in
span{U} and one orthogonal to U

1. Find the orthogonal projection of the matrix
[[3,2][4,5]] onto the space of diagonal 2x2 matrices of the form
lambda?I.
[[4.5,0][0,4.5]] [[5.5,0][0,5.5]] [[4,0][0,4]] [[3.5,0][0,3.5]] [[5,0][0,5]] [[1.5,0][0,1.5]]
2. Find the orthogonal projection of the matrix
[[2,1][2,6]] onto the space of symmetric 2x2 matrices of trace
0.
[[-1,3][3,1]] [[1.5,1][1,-1.5]] [[0,4][4,0]] [[3,3.5][3.5,-3]] [[0,1.5][1.5,0]] [[-2,1.5][1.5,2]] [[0.5,4.5][4.5,-0.5]] [[-1,6][6,1]] [[0,3.5][3.5,0]] [[-1.5,3.5][3.5,1.5]]
3. Find the orthogonal projection of the matrix
[[1,5][1,2]] onto the space of anti-symmetric 2x2
matrices.
[[0,-1] [1,0]] [[0,2] [-2,0]] [[0,-1.5]
[1.5,0]] [[0,2.5] [-2.5,0]] [[0,0]
[0,0]] [[0,-0.5] [0.5,0]] [[0,1] [-1,0]]
[[0,1.5] [-1.5,0]] [[0,-2.5]
[2.5,0]] [[0,0.5] [-0.5,0]]
4. Let p be the orthogonal projection of
u=[40,-9,91]T onto the...

1) Consider two vectors A=[20, 4, -6] and B=[8, -2, 6].
a) compute their dot product A.B
b) Compute the angle between the two vectors.
c)Find length and sign of component of A over B (mean Comp A
over B)and draw its diagram.
d) Compute Vector projection of B over A (means Proj B over A)
and draw corresponding diagram.
e) Compute Orthogonal projection of A onto B.

Let A be a given (3 × 3) matrix, and consider the equation Ax =
c, with c = [1 0 − 1 ]T . Suppose that the two vectors
x1 =[ 1 2 3]T and x2 =[ 3 2 1] T are
solutions to the above equation.
(a) Find a vector v in N (A).
(b) Using the result in part (a), find another solution to the
equation Ax = c.
(c) With the given information, what are the...

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