Linear Algebra Test
(1) Determine whether each matrix below is in echelon form, reduced echelon form, or neither of these.
(2) Given the system of equations
(a) Write down the corresponding augmented matrix
(b) Show explicitly the first few (around three) row operations . Stop when the
matrix is in echelon form.
(c) How many pivots will this matrix have?
(3) An system of linear equations and the corresponding row-reduced echelon form matrix appear below:
(a) Write down the general solution in the homogeneous case
(when b1 = b2 = b3 = 0)
(b) Is there a solution when b 1 = 3, b2 = 1, and b3 = 4 ?
(c) What must the relation be among the quantities b1, b2, and b3 in order
for this system of equations to be consistent ?
(4) True or False?
(a) If a system of linear equations has two different solutions , then it must
have infinitely many solutions.
(b) The equation 
has only the trivial
solution if and only if the row-reduced
matrix has a pivot in each row.
(5) For which value of h is the vector 
in
the set span 
(6) Determine whether each set of vectors below is linearly independent or
linearly
dependent:
(7) If a linear transformation is defined by
(a) What is the domain of T?
(b) What is the codomain of T?
(c) Is the column vector [1, 1, 1] in the range of T ?
(d) If the column vector 
is in the range of
T, what relation must
hold between the quantities b1 , b2 , and b3 ?
(e) Describe geometrically the range of the transformation T.
(f) Is the transformation T onto ?
(g) Is the transformation T one-to-one ?
(8) True or False?
(a) If 
is a linearly dependent set in R3,
then each vector is a
linear combination of the other two.
(b) If 
, then T is not a linear transformation.
(9) Find the standard matrix of the linear transformation
T : R2 -> R2 which
first reflects points across the diagonal line x2 = x1 , and then reflects
across
the horizontal axis x2 = 0.
(10) Write down a reduced echelon matrix for an augmented
matrix which represents
an overdetermined system of 5 homogeneous linear equations in 3 unknowns
with one free variable .
(11) Suppose that the matrix A on the left row- reduces to
the matrix B to the
right
Denote the columns of A by
, and the columns of B by
(a) True or False: The solution set of
is identical to the solution set
of 
(b) Is 
(c) Is span
(d) Do the columns of A form a linearly independent set?
(e) Is the set 
a linearly independent set?
(f) Is the linear transformation 
onto?
one-to-one ?
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