Let A be an m × n matrix, let x Rn and let 0 be the zero vector in Rm. (a) Let u, v є Rn be any two solutions of Ax 0,...
1. Let U с Rn be open, f : U-> Rm be a function, a є U and 0 exists. Show that DAwf(a) exists for every 0メλ R, and DAwf(a) Rn such that Duf(a) λDuf(a). 3 marks 1. Let U с Rn be open, f : U-> Rm be a function, a є U and 0 exists. Show that DAwf(a) exists for every 0メλ R, and DAwf(a) Rn such that Duf(a) λDuf(a). 3 marks
2. Given that u, v, and w are three solutions of the linear system Ax = b. Verify that the vector cu + dv + (1 − c − d)w is also a solution of Ax = b for any scalars c, d ∈ R.
Problem 5 (a) Let A be an n × m matrix, and suppose that there exists a m × n matrix B such that BA = 1- (i) Let b є Rn be such that the system of equations Ax b has at least one solution. Prove that this solution must be unique. (ii) Must it be the case that the system of equations Ax = b has a solution for every b? Prove or provide a counterexample. (b) Let...
Let U be an open subset of R. Let f: U C Rn → Rm. (a) Prove that f is continuously differentiable if and only if for each a є U, for each E > 0, there exists δ > 0 such that for each x E U, if IIx-all < δ, then llDf(x)-Df(a) ll < ε. (b) Let m n. Prove that if f is continuously differentiable, a E U, and Df (a) is invertible, then there exists δ...
2. Given that u, v, and w are three solutions of the linear system Ac = b. Verify that the vector cu + dv + (1 - C-d)w is also a solution of Ax b for any scalars c, d E R.
7. Let A be a 4 x 3 matrix, and let b and y be two arbitrary vectors in R. We are told that the system Ax- b has a unique solution. What can you say about the number of solutions of the system Ax - y? Explain your answer. 8. Let u. v, w, b be arbitrary vectors in R". Suppose that b = x1u+xy+23w for some scalars i, r23. Show that Span u, v, w, b Span u,...
3. Let U E Rnxn be an orthogonal matrix, i.c., UTU = UUT-1. Show that for any vector x E Rn LXTL we have |lU 2 2. Thus the 2-norm of a vector does not change when it is multiplied by an orthogonal matrix. 3. Let U E Rnxn be an orthogonal matrix, i.c., UTU = UUT-1. Show that for any vector x E Rn LXTL we have |lU 2 2. Thus the 2-norm of a vector does not change...
SOLVE BOTH 2 and 3 2. Given that u,v, and w are three solutions of the linear system Az = b. Verify that the vector cu + dv + (1-c-d)w is also a solution of Ax = b for any scalars c, d ER 3. Let A= -21 1 - 2 1 -2 Determine whether the system Ar = b is consistent for every beR'.
Given u 0 in Rn, let L-Spanu). For each y in Rh, the reflection of y in L is the point reflyy defined by reflLy 2 projy-y The figure shows that reflyy is the sum of proy andý -y Show that the mapping y- ref y is a linear transformation L = Span{u refly y The refiection of y in a line through the origin Let Ty)- refy2 proy-y. How can it be shown that T(y) is a linear transformation?...
a.) if A is an m*n matrix, such that Ax=0 for every vector x in R^n, then A is the m * n Zero matrix b.) The row echelon form of an invertible 3 * 3 matrix is invertible c.) If A is an m*n matrix and the equation Ax=0 has only the trivial solution, then the columns of A are linearly independent. d.) If T is the linear transformation whose standard matrix is an m*n matrix A and the...