True False Linear Algebra

Every elementary row operation is reversible : True

A 5x6 matrix has six rows : False: "If m and n are positive integers, an m x n matrix is a rectangular array of numbers w ith m row s and n columns."

Two equivalent linear systems can have different solution sets. : False: "Suppose a system is changed to a new one via row operations. By considering each type of row operation, you can see that any solution of the original system remains a solution of the new system.

A consistent system of linear equations has one or more solutions. : True: "A system of linear equations is said to be consistent if it has either one solution or infinitely many solutions;... "

If one row in an echelon form of an augmented matrix is [ 0 0 0 5 0] , then the associated linear system is inconsistent. : False. "A linear system is consistent if and only if the rightmost column of the augmented matrix is not a pivot column— that is, if and only if an echelon form of the augmented matrix has no row of the form [0 ... 0 b] with b nonzero."

If the equation Ax=b is consistent, then b is in the set spanned by the columns of A . : True. The equation Ax=b has a solution if and only if b is a linear combination of the columns of A .

The equation Ax=b is homogeneous if the zero vector is a solution. : True. Since A0=b=0, the equation Ax=b is also Ax=0 and thus is homogeneous.

The general solution of system is an explicit description of all solutions of the system. : True.

The solution set of the linear system w hose augmented matrix is [a1 a2 a3 b] is the same as the solution set of the equation x1a1 + x2a2 + x3a3 = b : True.

When u and v are nonzero vectors, Span {u, v} contains only the line through u and the origin, and the line through v and the origin. : False.

The column space of A is the set of all vectors that can be written as Ax for some x. : True.

Elementary row operations on an augmented matrix can change the solution set of the associated linear system. : False.

The effect of adding p to a vector is to move the vector in a direction parallel to p. : True.

The equation Ax=b is homogeneous if the zero vector is a solution. : True.

If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent. : False. The zero vector. Has two entries, not independent.

If a set in R^n is linearly dependent then the set contains more than n vectors. : False. The zero vector in R^2 is dependent but doesn't have more than 2 vectors.

A linear transformation T: R^n->R^m always maps the origin of R^n to the origin of R^m : True.

Every linear transformation from R^n to R^m is a matrix transformation. : True

For an mxn matrix Aand i<=i<=m, if x in the null space of A then x is orthogonal to Ai., the ith row of A . : True.

If the problem Ax=b has a solution x, then the problem HAx=Hb has the same solution x, for any matrix H. : True.

If the problem Ax=b has any solution x, then b must be in the column space of A : True

For any mxn matrix A, the null space of A is a subspace of R^m : False

If the columns of A are linearly dependent, then det(A)=0. : True.

det(A+B) = det(A)det(B) : False.