Question

Given that A and B are n × n matrices and T is a linear transformation. Determine which of the following is FALSE.

(a) If AB is not invertible, then either A or B is not invertible.

(b) If Au = Av and u and v are 2 distinct vectors, then A is not invertible.

(c) If A or B is not invertible, then AB is not invertible.

(d) If T is invertible and T(u) = T(v), then u = v.

(e) none of these

Given An×nx = 0 has only the trivial solution, determine which of the following is FALSE.

(a) Ax = 0 does not have any free variable and thus every column of A is a pivot column.

(b) Ax = b has exactly one solution A−1 b for every b ∈ Rn

(c) A is row equivalent to In.

(d) Every row of A has a pivot position.

(e) none of these

Given that columns of Bn×n are linearly dependent, determine which of the following is FALSE.

(a) Bx = 0 has nontrivial solutions and B is not invertible.

(b) Columns of Bn×n do not span Rn .

(c) Bx = b is inconsistent for some b.

(d) BT x = 0 has more than one solution.

(e) None of these

Answer #1

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...

n x n matrix A, where n >= 3. Select 3 statements from the
invertible matrix theorem below and show that all 3 statements are
true or false. Make sure to clearly explain and justify your
work.
A=
-1 , 7, 9
7 , 7, 10
-3, -6, -4
The equation A has only the trivial solution.
5. The columns of A form a linearly independent set.
6. The linear transformation x → Ax is one-to-one.
7. The equation Ax...

Suppose that A is an invertible n by
n matrix, with real valued entries.
Which of the following statements are true?
Select ALL correct answers.
Note: three submissions are allowed for this question.
A is row equivalent to the identity matrix
In.A has fewer than n pivot
positions.The equation
Ax=0 has only the
trivial solution.For some vector b in
Rn, the equation
Ax=b has no
solution.There is an n by n matrix C
such that CA=In.None of the above.

For each statement below, either show that the statement is true
or give an example showing that it is false. Assume throughout that
A and B are square matrices, unless otherwise specified.
(a) If AB = 0 and A ̸= 0, then B = 0.
(b) If x is a vector of unknowns, b is a constant column vector,
and Ax = b has no solution, then Ax = 0 has no solution.
(c) If x is a vector of...

4. Suppose that we have a linear system given in matrix form as
Ax = b, where A is an m×n matrix, b is an m×1 column vector, and x
is an n×1 column vector. Suppose also that the n × 1 vector u is a
solution to this linear system. Answer parts a. and b. below.
a. Suppose that the n × 1 vector h is a solution to the
homogeneous linear system Ax=0.
Showthenthatthevectory=u+hisasolutiontoAx=b.
b. Now, suppose that...

7. Answer the following questions true or false and provide an
explanation. • If you think the statement is true, refer to a
definition or theorem. • If false, give a counter-example to show
that the statement is not true for all cases.
(a) Let A be a 3 × 4 matrix. If A has a pivot on every row then
the equation Ax = b has a unique solution for all b in R^3 .
(b) If the augmented...

True or False
(5). Suppose the matrix A and B are both invertible, then (A +
B)−1 = A−1 + B−1
. (6). The linear system ATAx = ATb is always consistent for any
A ∈ Rm×n, b ∈Rm .
(7). For any matrix A ∈Rm×n , it satisﬁes dim(Nul(A)) =
n−rank(A).
(8). The two linear systems Ax = 0 and ATAx = 0 have the same
solution set.
(9). Suppose Q ∈Rn×n is an orthogonal matrix, then the row...

True or False? No reasons needed.
(e) Suppose β and γ are bases of F n and F m, respectively.
Every m × n matrix A is equal to [T] γ β for some linear
transformation T: F n → F m.
(f) Recall that P(R) is the vector space of all polynomials with
coefficients in R. If a linear transformation T: P(R) → P(R) is
one-to-one, then T is also onto.
(g) The vector spaces R 5 and P4(R)...

Let E be an n×n matrix, and letU= {xE:x∈Rn} (where x∈Rn is
written as arow vector). Show that the following are
equivalent.
(a) E^2 = E = E^T (T means transpose).
(b) (u − uE) · (vE) = 0 for all u, v ∈ Rn.
(c) projU(v) = vE for all v ∈ Rn.

True/ false
a- If the last row in an REF of an augmented matrix is [0 0 0 4
0], then the associated linear system is inconsistent.
b-The equation Ax=b is consistent if the augmented matrix [A b]
has a pivot position in every row.
c-The set Span{v} for a nonzero v is always a line that may or
may not pass through the origin.

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