Suppose A is an n × n matrix with the property that the equation Ax =...

Suppose A is an n × n matrix with the property that the equation Ax = b has at least one solution for each b in R n . Explain why each equation Ax = b has in fact exactly one solution

Expert Solution

Colby Messinger answered 1 year ago

for the system equation of x' = Ax if Coefficients Matrix A be ? = [...

for the system equation of x' = Ax if Coefficients Matrix A be ? = [ 5 −5 −5 −1 4 2 3 −5 −3 ] , find the basic matrix

Suppose the system AX = B is consistent and A is a 6x3 matrix. Suppose the...

Suppose the system AX = B is consistent and A is a 6x3 matrix. Suppose the maximum number of linearly independent rows in A is 3. Discuss: Is the solution of the system unique?

4. The product y = Ax of an m × n matrix A times a vector...

4. The product y = Ax of an m × n matrix A times a vector x = (x1, x2, . . . , xn) T can be computed row-wise as y = [A(1,:)*x; A(2,:)*x; ... ;A(m,:)*x]; that is y(1) = A(1,:)*x y(2) = A(2,:)*x ... y(m) = A(m,:)*x Write a function M-file that takes as input a matrix A and a vector x, and as output gives the product y = Ax by row, as defined above (Hint: use...

Suppose C is a m × n matrix and A is a n × m matrix....

Suppose C is a m × n matrix and A is a n × m matrix. Assume CA = Im (Im is the m × m identity matrix). Consider the n × m system Ax = b. 1. Show that if this system is consistent then the solution is unique. 2. If C = [0 ?5 1 3 0 ?1] and A = [2 ?3 1 ?2 6 10] ,, find x (if it exists) when (a) b =[1...

(a) The n × n matrices A, B, C, and X satisfy the equation AX(B +...

(a) The n × n matrices A, B, C, and X satisfy the equation AX(B + CX) ?1 = C Write an expression for the matrix X in terms of A, B, and C. You may assume invertibility of any matrix when necessary. (b) Suppose D is a 3 × 5 matrix, E is a 5 × c matrix, and F is a 4 × d matrix. Find the values of c and d for which the statement “det(DEF) =...

4. The product y = Ax of an m n matrix A times a vector x...

4. The product y = Ax of an m n matrix A times a vector x = (x1; x2; : : : ; xn)T can be computed row-wise as y = [A(1,:)*x; A(2,:)*x; ... ;A(m,:)*x]; that is y(1) = A(1,:)*x y(2) = A(2,:)*x ... y(m) = A(m,:)*x Write a function M-file that takes as input a matrix A and a vector x, and as output gives the product y = Ax by row, as denoted above (Hint: use a for...

Let A be an m x n matrix. Prove that Ax = b has at least...

Let A be an m x n matrix. Prove that Ax = b has at least one solution for any b if and only if A has linearly independent rows. Let V be a vector space with dimension 3, and let V = span(u, v, w). Prove that u, v, w are linearly independent (in other words, you are being asked to show that u, v, w form a basis for V)

Suppose that A is an n × n matrix satisfying A3 - 3A + 2A-3l =...

Suppose that A is an n × n matrix satisfying A3 - 3A + 2A-3l = 0. Show that A is invertible by the definition of invertible. (Hint: Review the definition of invertible, and then describe the inverse in terms of the matrix A - you don’t need to know what A is to answer this question.)

Problem 3. Throughout this problem, we fix a matrix A ∈ Fn,n with the property that...

Problem 3. Throughout this problem, we fix a matrix A ∈ Fn,n with the property that A = A∗. (If F = R, then A is called symmetric. If F = C, then A is called Hermitian.) For u, v ∈ Fn,1, define [u, v] = v∗ Au. (a) Let Show that K is a subspace of Fn,1. K:={u∈Fn,1 :[u,v]=0forallv∈Fn,1}. (b) Suppose X is a subspace of Fn,1 with the property that [v,v] > 0 for all nonzero v ∈...

What property must a symmetric 3 × 3 matrix have in order for the equation xTAx...

What property must a symmetric 3 × 3 matrix have in order for the equation xTAx = 1 to represent an ellipsoid? (6 points)(Kindly provide a long, comprehensive proof)

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