(a) Define the Range and Null spaces of a matrix AR*, respectively denoted by R[A] and...

  

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(a) Define the Range and Null spaces of a matrix AR*, respectively denoted by R[A] and N [A]. Prove that they are both linear vector spaces. What are their dimensions? Show that the range of a matrix is the linear vector space span {m} of its columns c, i = 1,...,m. (b) Consider the solution of a linear problem Ary, rR", yER", AER** (4) with respect to r, when A and y are known. Prove that problem (??) can be re-stated as a mini - mization problem for a standard quadratic Derive conditions under which it has an exact solution. Do your conditions guarantee that such an exact solution is unique ? (c) Provide the definition of an orthogonal plement unique ? Provide a geometric to the solution of (??) in the case when complement of a subspace SCR". Is the orthogonal com- interpretation of the calculation of the best approximation y R[A]. Prove that if is such a best approximation then (y A) ER[A] - Provide a graphical interpretation of your proof. (5) Q.5 (a) Prove that = N[A] C. (1)] where the i=1,...,n are the last n - r columns of the matrix Q in the orthogonal decompo sition of A: A = P (6) (7) Explain how this result can be used to find a matrix H such that R[H] = N [A]. (b) Prove that if a subspace CC R" is such that C = R[A] for some matrix AR orthogonal projections of a vector R": and 2, onto and its orthogonal complement respectively, are given by the formulae then the +C. 2 = 11 z = (1-AA) (8) (c) Prove that for any matrix A: RA] = NIA"] and explain how this result can be useful in calcula- tion of the pseudo-inverse At. This proof can be found in the notes, but please, copy it here with understanding Prove the similar statements (a) (b) (c) +R[AT]=N[A] R[AT]N[A] R[A] NA = (9) (10) (11) (a) Define the Range and Null spaces of a matrix AR*, respectively denoted by R[A] and N [A]. Prove that they are both linear vector spaces. What are their dimensions? Show that the range of a matrix is the linear vector space span {m} of its columns c, i = 1,...,m. (b) Consider the solution of a linear problem Ary, rR", yER", AER** (4) with respect to r, when A and y are known. Prove that problem (??) can be re-stated as a mini - mization problem for a standard quadratic Derive conditions under which it has an exact solution. Do your conditions guarantee that such an exact solution is unique ? (c) Provide the definition of an orthogonal plement unique ? Provide a geometric to the solution of (??) in the case when complement of a subspace SCR". Is the orthogonal com- interpretation of the calculation of the best approximation y R[A]. Prove that if is such a best approximation then (y A) ER[A] - Provide a graphical interpretation of your proof. (5) Q.5 (a) Prove that = N[A] C. (1)] where the i=1,...,n are the last n - r columns of the matrix Q in the orthogonal decompo sition of A: A = P (6) (7) Explain how this result can be used to find a matrix H such that R[H] = N [A]. (b) Prove that if a subspace CC R" is such that C = R[A] for some matrix AR orthogonal projections of a vector R": and 2, onto and its orthogonal complement respectively, are given by the formulae then the +C. 2 = 11 z = (1-AA) (8) (c) Prove that for any matrix A: RA] = NIA"] and explain how this result can be useful in calcula- tion of the pseudo-inverse At. This proof can be found in the notes, but please, copy it here with understanding Prove the similar statements (a) (b) (c) +R[AT]=N[A] R[AT]N[A] R[A] NA = (9) (10) (11)