Question

Let A be a given (3 × 3) matrix, and consider the equation Ax =
c, with c = [1 0 − 1 ]^{T} . Suppose that the two vectors
x1 =[ 1 2 3]^{T} and x2 =[ 3 2 1] ^{T} are
solutions to the above equation.

(a) Find a vector v in N (A).

(b) Using the result in part (a), find another solution to the equation Ax = c.

(c) With the given information, what are the possible values for null(A)?

Answer #1

Find the matrix A in the linear transformation y =
Ax,where a point x = [x1,x2]^T is projected on the x2 axis.That
is,a point x = [x1,x2]^T is projected on to [0,x2]^T . Is A an
orthogonal matrix ?I any case,find the eigen values and eigen
vectors of A .

Let A be an m×n matrix, x a vector in Rn, and b a vector in Rm.
Show that if x1 in Rn is a solution to Ax=b and x2 is a solution to
Ax=⃗0, then x1 +x2 is a solution to Ax=b.

Given a matrix A, the equation Ax=b might have 0 solutions for
all b, or 1 solution for all b, or 0 solutions for some choices of
b and 1 solution for others, or 0 solutions for
some choices of b and ∞ solutions for others, or 1 solution for
some b and ∞ solutions for others, or 0,1, or ∞ solutions for
different choices of b. (7 different combinations in
all.) Which of these combinations are actually possible? Justify...

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?

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...

Let ? be an ?×? matrix, and consider the equation ??⃗ = 0⃗. Show
that for any ?⃗ ∈ ℝ?, the solution of the equation ??⃗ =
0⃗ which minimizes the distance between ?⃗ and ?⃗ is given by ?⃗ =
?⃗ − ???⃗ where ????⃗ = ??⃗ .

Let V be the subspace of all vectors in R 5 , such that x1 − x4
= x2 − 5x5 = 3x3 + x4
(a) Find a matrix A with that space as its Null space; What is
the rank of A?
b) Find a basis B1 of V ; What is the dimension of V ?
(c) Find a matrix D with V as its column space. What is the rank
of D? To find the rank of...

Consider a plane and a line given by equation ax +
by = 0. Write a 2×2 matrix, M, of the mirror
reflection (flip) around this line. Calculate det(M) and
M2 and discuss the results.

Find the fundamental matrix solution for the system x′ = Ax
where matrix A is given. If an initial condition is provided, find
the solution of the initial value problem using the principal
matrix.
A= [ 4 -13 ; 2 -6 ]. , x(o) = [ 2 ; 0 ]

2X1-X2+X3+7X4=0
-1X1-2X2-3X3-11X4=0
-1X1+4X2+3X3+7X4=0
a. Find the reduced row - echelon form of the coefficient
matrix
b. State the solutions for variables X1,X2,X3,X4 (including
parameters s and t)
c. Find two solution vectors u and v such that the solution
space is \
a set of all linear combinations of the form su + tv.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 16 minutes ago

asked 19 minutes ago

asked 25 minutes ago

asked 27 minutes ago

asked 34 minutes ago

asked 35 minutes ago

asked 35 minutes ago

asked 44 minutes ago

asked 53 minutes ago

asked 58 minutes ago

asked 1 hour ago

asked 1 hour ago