Question

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

Answer #1

(C) yes, No, Yes is correct answer

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...

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...

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)

Let B = [ aij ] 20×17 be a matrix with real entries. Let x be in
R 17 , c be in R 20, and 0 be the vector with all zero entries.
Show that each of the following statements implies the other.
(a) Bx = 0 has only the trivial solution x = 0 n R 17, then (b)
If Bx = c has a solution for some vector c in R 20, then the
solution is unique.

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...

1. Let a,b,c,d be row vectors and form the matrix A whose rows
are a,b,c,d. If by a sequence of row operations applied to A we
reach a matrix whose last row is 0 (all entries are 0) then:
a. a,b,c,d are linearly dependent
b. one of a,b,c,d must be 0.
c. {a,b,c,d} is linearly independent.
d. {a,b,c,d} is a basis.
2. Suppose a, b, c, d are vectors in R4 . Then they form a...

Here are some vectors in R 4 : u1 = [1 3 −1 1] u2 = [1 4 −1 1]
u3 = [1 0 −1 1] u4 = [2 −1 −2 2] u5 = [1 4 0 1]
(a) Explain why these vectors cannot possibly be
independent.
(b) Form a matrix A whose columns are the ui’s and compute the
rref(A).
(c) Solve the homogeneous system Ax = 0 in parametric form and
then in vector form. (Be sure the...

A linear system of equations Ax=b is known, where A is a matrix
of m by n size, and the column vectors of A are linearly
independent of each other. Please answer the following questions
based on this assumption, please explain it, thank you~.
(1) To give an example, Ax=b is the only solution.
(2) According to the previous question, what kind of inference
can be made to the size of A at this time? (What is the size of...

a)Assume that you are given a matrix A = [aij ] ∈ R n×n with (1
≤ i, j ≤ n) and having the following interesting property:
ai1 + ai2 + ..... + ain = 0 for each i = 1, 2, ...., n
Based on this information, prove that rank(A) < n.
b) Let A ∈ R m×n be a matrix of rank r. Suppose there are right
hand sides b for which Ax = b has no solution,...

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