Question

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?

Answer #1

Linear Algebra
Find the 2x2 matrix A of a linear transformation T: R^2->R^2
such that T(vi) = wi for i = 1,2.
v1=(4,3), v2=(5,4); w1=(-3,2), w2=(-3,-4)

(a) Prove that if two linear transformations T,U : V --> W
have the same values on a basis for V, i.e., T(x) = U(x) for all x
belong to beta , then T = U. Conclude that every linear
transformation is uniquely determined by the images of basis
vectors.
(b) (7 points) Determine the linear transformation T : P1(R)
--> P2(R) given by T (1 + x) = 1+x^2, T(1- x) = x by finding the
image T(a+bx) of...

#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...

find a 4×4 matrix B such that the linear transformation x⇝Bx
preserves distance but does not preserve orientation

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)

Find Eigenvalues and Eigenspaces for matrix:
The 2 × 2 matrix AT associated to the linear transformation T :
R2 → R2 which rotates a vector π/4-radians then reflects it about
the x-axis.

Linear algebra
Find a transformation matrix A for P^2 -> P^3 then show its
onto

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

Find the standard matrix for the following transformation T : R
4 → R 3 : T(x1, x2, x3, x4) = (x1 − x2 + x3 − 3x4, x1 − x2 + 2x3 +
4x4, 2x1 − 2x2 + x3 + 5x4) (a) Compute T(~e1), T(~e2), T(~e3), and
T(~e4). (b) Find an equation in vector form for the set of vectors
~x ∈ R 4 such that T(~x) = (−1, −4, 1). (c) What is the range of
T?

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.

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