Let b be a vector that has the same height as a square matrix A. How...

Let b be a vector that has the same height as a square matrix A. How many solutions can Ax = b have?

Expert Solution

Colby Messinger answered 2 years ago

Let vector B = 5.50 m at 60°. Let vector C have the same magnitude as...

Let vector B = 5.50 m at 60°. Let vector C have the same magnitude as vector A and a direction angle greater than that of vector A by 25°. Let vector A · vector B = 27.0 m2 and vector B · vector C = 34.5 m2. Find the magnitude and direction of vector A.

a. Let A be a square matrix with integer entries. Prove that if lambda is a...

a. Let A be a square matrix with integer entries. Prove that if lambda is a rational eigenvalue of A then in fact lambda is an integer. b. Prove that the characteristic polynomial of the companion matrix of a monic polynomial f(t) equals f(t).

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)).

Let [x]B be the coordinate vector of a vector x ∈ V with respect to the...

Let [x]B be the coordinate vector of a vector x ∈ V with respect to the basis B for V . Show that x is nonzero if and only if [x]B is nonzero.

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)

Let A ∈ Mat n×n(R) be a real square matrix. (a) Suppose that A is symmetric,...

Let A ∈ Mat n×n(R) be a real square matrix. (a) Suppose that A is symmetric, positive semi-definite, and orthogonal. Prove that A is the identity matrix. (b) Suppose that A satisfies A = −A^T . Prove that if λ ∈ C is an eigenvalue of A, then λ¯ = −λ. From now on, we assume that A is idempotent, i.e. A^2 = A. (c) Prove that if λ is an eigenvalue of A, then λ is equal to 0...

Prove uniqueness of (a) LU-factorisation, (b) of LDU-factorisation of a square matrix.

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 B equal the matrix below: [{1,0,0},{0,3,2},{2,-2,-1}] (1,0,0 is the first row of the matrix B)...

Let B equal the matrix below: [{1,0,0},{0,3,2},{2,-2,-1}] (1,0,0 is the first row of the matrix B) (0,3,2 is the 2nd row of the matrix B (2,-2,-1 is the third row of the matrix B.) 1) Determine the eigenvalues and associated eigenvectors of B. State both the algebraic and geometric multiplicity of the eigenvalues. 2) The matrix is defective. Nonetheless, find the general solution to the system x’ = Bx. (x is a vector)

Let Vand W be vector spaces over F, and let B( V, W) be the set...

Let Vand W be vector spaces over F, and let B( V, W) be the set of all bilinear forms f: V x W ~ F. Show that B( V, W) is a subspace of the vector space of functions 31'( V x W). Prove that the dual space B( V, W)* satisfies the definition of tensor product, with respect to the bilinear mapping b: V x W -> B( V, W)* defined by b(v, w)(f) =f(v, w), f E...

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