LA LA MIDTERM

Invertible matrix thmLet A be a square n x n matrixthen the following statements are equivalent. That is, for a given A, the statements are either all true or all false : a. A is an invertible matrixb. A is row equivalent to the n x n identity matrixc. A has n pivot positions d. the equation ax = 0 has only trivial solne. columns of A form linearly independent setf. linear transform x -> Ax is one-to-oneg. equation Ax = b has at least one soln for each b in R^nh. columns of A span R^ni. the linear transform x-> Ax maps R^n onto R^nj. there is an n x n matrix C such that CA = Ik. there is an n x n matrix D such that AD = Il. A^T is an invertible matrix

A system of linear equations with fewer equations than unknowns is sometimes called an underdetermined system. Can such a system have a unique solution? : An underdetermined system always has more variables than equations.There cannot be more basic variables than there are equations, so there must be at least one free variable. If the system is consistent, each different value of a free variable will produce a different solution and the system will not have a unique soln. If the system is inconsistent, it will not have any soln.

Explain why the columns of an n x n matrix A are linearly independent when A is invertible : Suppose A is invertible. By Thm 5, the equation Ax = 0 has only one soln, namely, the zero soln. This means the columns of A are linearly independent

What is the rank of a 6 x 8 matrix whose null space is three dimensional : A 6 x 8 matrix A has 8 columns. By the Rank Thm, rank A = 8 - dim Nul A. Since the null space is three dimensinoal, rank A = 5

T/FIf a set of p vectors spans a p-dimensinal subspace H of R^n, then these vectors form a basis for H : True. by the Basis Thm In this case, the spanning set is automaticaly a linearly independent set

T/FIf B is a basis for a subspace H, then each vector in H can be written in only one way as a linear combination of the vectors in B : True

Suppose a 4 x 6 matrix A has four pivot columns. Is Col A = R^4? Is Nul A = R^2? : Col A = R^4 b/c A has a pivot in each row and so the columns of A span R^4Nul A cannot equal R^2 because Nul A is a subspace of R^6It is true, however, that Nul A is two-dimensional. Reason: the equation Ax = 0 has two free variables because A has six columns and only four of them are pivot columns

What is the rank of a 6 x 8 matrix whose null space is 3-dimensinal? : A 6 x 8 matrix A has 8 columns. By rank thm, rank A = 8 - dim Nul A. Since the null space is three dimensional, rank A = 5