Question

Answer all of the questions true or false:
1.
a) If one row in an echelon form for an augmented matrix is [0 0 5 0 0]
b) A vector b is a linear combination of the columns of a matrix A if and only if the
equation Ax=b has at least one solution.
c) The solution set of b is the set of all vectors of the form u = + p + vh
where vh is any solution of the equation Ax = 0 and is a solution of Ax=b.
d) The columns of a matrix A are linearly independent if the equation = O has
the trivial solution.
e)lf A and B are invertible nxn matrices then A^-1 B^-1 is the inverse of AB.

Answer all of the questions true or false:

2.
a) Two matrices are row equivalent if they have the same number of rows.
b) The reduced echelon form of a matrix is unique.
c) The weights c1, c2, ... , cp on a linear combination +
+ cpvp cannot all be zero.
d) If the equation Ax = b is consistent then b is in the span of the columns of A.
e) A linear transformation preserves the operations of vector addition and scalar
multiplication.

Answer all of the questions true or false:
3.
a) The determinant of a triangular matrix is the sum of the diagonal entries.
b) If the columns of A are linearly dependent then det A = 0.
c) The columns of an nxn invertible matrix form a basis for Rn.
d) The dimension of the vector space P4 is 4.
e) The dimension of the row space and column space of A is the same, even if A is
not square

Answer all of the questions true or false:
4.
a) If the columns of A are linearly dependent then detA = 0.
b) R2 is a subspace of R3.
c) The null space of A is the solution set for the equation Ax = 0.
d) If is in V and if B contains n vectors then the B-coordinate vector of is in
e) If B is an echelon form of A then the pivot columns of B form a basis for the
column space of A.

Answer all of the questions true or false:

5.
a) If = for some then is an eigenvalue for A.
b) The eigenvalues of A are the entries on its diagonal.
c) If A is diagonaizable then A is invertible.
d) Not every linearly independent set is an an othorgonal set.
e) If ||u+v||^2 = ||u||^2 + ||v||^2 then u and v are orthogonal.

Answer all of the questions true or false:

6.
a) To find an eigenvalue for A, reduce A to echelon form.
b) If A is invertible then A is diagonalizable.

c) For any scalar c, ||cu|| = c||u||.

d) If the columns of A are orthonormal, then the linear transofrmation c -> Ax perserves length.

e) If A is orthogonal then A is invertible.

Answer 1