Question

Let *A* be an *n* by *n* matrix, with real
valued entries. Suppose that *A* is NOT invertible.

Which of the following statements are true?

*?Select ALL correct answers.?*

The columns of *A* are linearly dependent.

The linear transformation given by *A* is one-to-one.

The columns of *A* span
**R*** ^{n}*.

The linear transformation given by *A* is onto
**R*** ^{n}*.

There is no *n* by *n* matrix *D* such that
*AD*=*I** _{n}*.

None of the above.

Answer #1

et A be an n by n matrix, with real
valued entries. Suppose that A is NOT invertible.
Which of the following statements are true?
Select ALL correct answers.
Note: three submissions are allowed for this question.
The columns of A are linearly independent.The linear
transformation given by A is not one-to-one.The columns of
A span Rn.The linear
transformation given by A is onto
Rn.There is an n by
n matrix D such that
AD=In.None of the above.

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.

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 =...

Let
A be an n by n matrix. Prove that if the linear transformation L_A
from F^n to F^n defined by L_A(v)=Av is invertible then A is
invertible.

Let A be an m × n matrix with m ≥ n and linearly independent
columns. Show that if z1, z2, . . . , zk is a set of linearly
independent vectors in Rn, then Az1,Az2,...,Azk are linearly
independent vectors in Rm.

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...

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)...

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...

Let X be a real vector space. Suppose {⃗v1,⃗v2,⃗v3} ⊂ X is a
linearly independent set, and suppose {w⃗1,w⃗2,w⃗3} ⊂ X is a
linearly dependent set. Define V = span{⃗v1,⃗v2,⃗v3} and W =
span{w⃗1,w⃗2,w⃗3}.
(a) Is there a linear transformation P : V → W such that P(⃗vi)
= w⃗i for i = 1, 2, 3?
(b) Is there a linear transformation Q : W → V such that Q(w⃗i)
= ⃗vi for i = 1, 2, 3?
Hint: the...

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