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)

Expert Solution

Colby Messinger answered 1 year ago

Let A be an m x n matrix and b and x be vectors such that...

Let A be an m x n matrix and b and x be vectors such that Ab=x. a) What vector space is x in? b) What vector space is b in? c) Show that x is a linear combination of the columns of A. d) Let x' be a linear combination of the columns of A. Show that there is a vector b' so that Ab' = x'.

Let A be an m × n matrix and B be an m × p matrix....

Let A be an m × n matrix and B be an m × p matrix. Let C =[A | B] be an m×(n + p) matrix. (a) Show that R(C) = R(A) + R(B), where R(·) denotes the range of a matrix. (b) Show that rank(C) = rank(A) + rank(B)−dim(R(A)∩R(B)).

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...

Problem 1 1.1 If A is an n x n matrix, prove that if A has...

Problem 1 1.1 If A is an n x n matrix, prove that if A has n linearly independent eigenvalues, then AT is diagonalizable. 1.2 Diagonalize the matrix below with eigenvalues equal to -1 and 5. 0 1 1 2 1 2 3 3 2 1.3 Assume that A is 4 x 4 and has three different eigenvalues, if one of the eigenspaces is dimension 1 while the other is dimension 2, can A be undiagonalizable? Explain. Answer for all...

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...

let n belongs to N and let a, b belong to Z. prove that a is...

let n belongs to N and let a, b belong to Z. prove that a is congruent to b, mod n, if and only if a and b have the same remainder when divided by n.

(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) =...

A linear system of equations Ax=b is known, where A is a matrix of m by...

A linear system of equations Ax=b is known, where A is a matrix of m by n size, and the column vectors of A are linearly independent of each other. Please answer the following questions based on this assumption, please explain all questions, thank you~. (1) Please explain why this system has at most one solution. (2) To give an example, Ax=b is no solution. (3) According to the previous question, what kind of inference can be made to the...

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

Prove the trace of an n x n matrix is an element of the dual space...

Prove the trace of an n x n matrix is an element of the dual space of all n x n matrices.

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