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Set

A = rand(8, 4) * rand(4, 6), [U, D, V]= svd(A)

(a) What is the rank of A? Use the column vectors of V to generate two matrices V l and V2 whose columns form orthonormal bases for R(AT) and N(A), respectively. Set

p = V2 * V2ʹ, r = P * rand(6, 1), w = Aʹ * rand(8, 1)

If r and w had been computed in exact arithmetic, they would be orthogonal. Why? Explain. Use MATLAB to compute rTw.

(b) Use the column vectors of U to generate two matrices U1 and U2 whose column vectors form orthonormal bases for R(A) and N(AT), respectively. Set

Q = U2*U2ʹ, y = Q* rand(8. 1), z = A * rand(6, 1)

Explain why y and z would be orthogonal if all computations were done in exact arithmetic. Use MATLAB to compute yTz.

(c) Set X = pinv(A). Use MATLAB to verify the four Penrose conditions:

(i) AXA = A

(ii) XAX = X

(iii) (AX)T = AX

(iv) (XA)T = XA

A = rand(8, 4) * rand(4, 6), [U, D, V]= svd(A)

(a) What is the rank of A? Use the column vectors of V to generate two matrices V l and V2 whose columns form orthonormal bases for R(AT) and N(A), respectively. Set

p = V2 * V2ʹ, r = P * rand(6, 1), w = Aʹ * rand(8, 1)

If r and w had been computed in exact arithmetic, they would be orthogonal. Why? Explain. Use MATLAB to compute rTw.

(b) Use the column vectors of U to generate two matrices U1 and U2 whose column vectors form orthonormal bases for R(A) and N(AT), respectively. Set

Q = U2*U2ʹ, y = Q* rand(8. 1), z = A * rand(6, 1)

Explain why y and z would be orthogonal if all computations were done in exact arithmetic. Use MATLAB to compute yTz.

(c) Set X = pinv(A). Use MATLAB to verify the four Penrose conditions:

(i) AXA = A

(ii) XAX = X

(iii) (AX)T = AX

(iv) (XA)T = XA

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