Question

Consider the matrix A=

−2−2 6]

[−2−3 5]

[3 4−8]

[−7−9 18

(all one matrix)

(a) How many rows ofAcontain a pivot position?

(b) Do the columns ofAspanR4?

(c) Does the equationA ~x=~b have a solution for every~b∈R^4?

(d) Would the equation A~x=~0 have a nontrivial solution?

(e) Are the columns of A linearly independent?

(~x is vector x)

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...

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...

Let A be a real matrix of 7 × 5 format. Answer the questions
following:
(1) Can the homogeneous system AX = 0 have a non-trivial solution?
(2) Can the columns of A form a generating system of R^7?
(3) Can the columns of A be linearly independent in R^7?
A. Yes, No, No D. No, No, Yes
B. Yes, Yes, Yes E. No, No, No
C. Yes, No, Yes F. No, Yes, Yes

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...

LINEAR ALGEBRA
For the matrix B=
1 -4 7 -5
0 1 -4 3
2 -6 6 -4
Find all x in R^4 that are mapped into the zero vector by the
transformation Bx.
Does the vector:
1
0
2
belong to the range of T? If it does, what is the pre-image of
this vector?

Refer to the following matrices.
A =
1
−8
7
−3
−18
6
5
8
6
0
5
8
3
4
8
−2
B =
4
−7
4
0
1
4
9
5
3
−3
0
9
C =
1
0
3
2
8
D =
1
9
−7
0
What is the size of each matrix?
A
____× ____
B
_____× _____
C
____× ____
D
____× _____

Let p = (8, 10, 3, 11, 4, 0, 5, 1, 6, 2, 7, 9) and let q = (2,
4, 9, 5, 10, 6, 11, 7, 0, 8, 1, 3) be tone rows. Verify that p =
Tk(R(I(q))) for some k, and find this value of k.

#2. For the matrix A = 1 2 1 2 3 7 4 7 9 find the
following. (a) The null space N (A) and a basis for N (A). (b) The
range space R(AT ) and a basis for R(AT )
. #3. Consider the vectors −→x = k − 6 2k 1 and −→y =
2k 3 4 . Find the number k such that the vectors...

Matrix A= -2 1 0
2 -3 4
5 -6 7
vector u= 1
2
1
a) Is the vector u in Null(A) Explain in detail why
b) Is the vectro u in Col( A) Explain in detail why

Consider the following data: 6, 8, 2, 3, 4, 4, 5, 5, 9, 6, 6, 7,
7 ,7, 7, 8, 8, 8, 8, 9, 9.
What type of distribution is this?

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