Question

8. Consider a 4 × 2 matrix A and a 2 × 5 matrix B .

(a) What are the possible dimensions of the null space of AB? Justify your answer.

(b) What are the possible dimensions of the range of AB Justify your answer.

(c) Can the linear transformation define by A be one to one? Justify your answer.

(d) Can the linear transformation define by B be onto? Justify your answer.

Answer #1

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?

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...

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)

If the matrix A is 4×2, B is 3×4, C is 2×4, D is 4×3,
and E is 2×5, what are the
dimensions of the following expressions
a) A^TD + CB^T
b) (B + D^T ) A
c) CA + CB^T

Recall that if A is an m × n matrix and B is a p × q matrix,
then the product C = AB is defined if and only if n = p, in which
case C is an m × q matrix. (a) Write a function M-file that takes
as input two matrices A and B, and as output produces the product
by columns of the two matrix. For instance, if A is 3 × 4 and B is...

Consider a transformation T : R2x2 -> R2x2 such that T(M) =
MT . This is in fact a linear transformation. Based on this,
justify if the following
statements are true or not.
a) T . T is the identity transformation.
b) The kernel of T is the zero matrix.
c) Range T = R2x2
d) T(M) =-M is impossible.

Suppose T:ℝ4→ℝ4 is the transformation
induced by the following matrix A. Determine whether T is
one-to-one and/or onto. If it is not one-to-one, show this by
providing two vectors that have the same image under T. If T is not
onto, show this by providing a vector in ℝ4 that is not
in the range of T.
A =
2
−2
2
−2
2
0
0
−10
2
−1
4
−9
2
−1
3
−8
T is one-to-one
T is...

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

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)...

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