Question

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 matrix [ A b ] has a non-pivot column then the system Ax = b has infinitely many solutions.

(c) If a1 and a2 are vectors in R^3 such that a1 is not a scalar multiple of a2, then {a1, a2} is an linearly independent set.

(d) Let A be a 6 × 4 matrix. If the columns of A are linearly independent then the columns of A span R^6 .

(e) Let T : R^3 → R^3 be a linear transformation. If T is onto, then T must also be one-to-one.

(f) Let a1, a2 and a3 be vectors in R^3 . If there exists real numbers x1, x2, x3 such that x1a1 + x2a2 + x3a3 = 0 then {a1, a2, a3} is a linearly dependent set.

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...

For each of the following statements, say whether the statement
is true or false.
(a) If S⊆T are sets of vectors, then span(S)⊆span(T).
(b) If S⊆T are sets of vectors, and S is linearly independent,
then so is T.
(c) Every set of vectors is a subset of a basis.
(d) If S is a linearly independent set of vectors, and u is a
vector not in the span of S, then S∪{u} is linearly
independent.
(e) In a finite-dimensional...

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.

Mark the following as true or false, as the case may be. If a
statement is true, then prove it. If a statement is false, then
provide a counter-example.
a) A set containing a single vector is linearly independent
b) The set of vectors {v, kv} is linearly dependent for every
scalar k
c) every linearly dependent set contains the zero vector
d) The functions f1 and f2 are linearly
dependent is there is a real number x, so that...

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...

Answer the following T for True and F for False:
__ A vector space must have an infinite number of vectors to be
a vector space.
__ The dimension of a vector space is the number of linearly
independent vectors contained in the vector space.
__ If a set of vectors is not linearly independent, the set is
linearly dependent.
__ Adding the zero vector to a set of linearly independent
vectors makes them linearly dependent.

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

k- If a and b are linearly independent, and if {a , b , c} is
linearly dependent, then c is in Span{a , b}.
Group of answer choices
j- If A is a 4 × 3 matrix, then the transformation described by
A cannot be one-to-one. true/ false
L-
If A is a 5 × 4 matrix, then the transformation x ↦ A x cannot
map R 4 onto R 5.
True / false

Consider an axiomatic system that consists of elements in a set
S and a set P of pairings of elements (a, b) that satisfy the
following axioms:
A1 If (a, b) is in P, then (b, a) is not in P.
A2 If (a, b) is in P and (b, c) is in P, then (a, c) is in
P.
Given two models of the system, answer the questions below.
M1: S= {1, 2, 3, 4}, P= {(1, 2), (2,...

Problem 2. (20 pts.) show that T is a linear transformation by
finding a matrix that implements the mapping. Note that x1, x2, ...
are not vectors but are entries in vectors. (a) T(x1, x2, x3, x4) =
(0, x1 + x2, x2 + x3, x3 + x4) (b) T(x1, x2, x3, x4) = 2x1 + 3x3 −
4x4 (T : R 4 → R)
Problem 3. (20 pts.) Which of the following statements are true
about the transformation matrix...

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