Question

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.

Answer #1

Let E be an n×n matrix, and letU= {xE:x∈Rn} (where x∈Rn is
written as arow vector). Show that the following are
equivalent.
(a) E^2 = E = E^T (T means transpose).
(b) (u − uE) · (vE) = 0 for all u, v ∈ Rn.
(c) projU(v) = vE for all v ∈ Rn.

Let A be an m × n matrix such that ker(A) = {⃗0} and let
⃗v1,⃗v2,...,⃗vq be linearly independent vectors in Rn. Show that
A⃗v1, A⃗v2, . . . , A⃗vq are linearly independent vectors in
Rm.

Problem 3.2
Let H ∈ Rn×n be symmetric and idempotent, hence a projection
matrix. Let x ∼ N(0,In). (a) Let σ > 0 be a positive number.
Find the distribution of σx. (b) Let u = Hx and v = (I −H)x and ﬁnd
the joint distribution of (u,v). 1 (c) Someone claims that u and v
are independent. Is that true? (d) Let µ ∈ Im(H). Show that Hµ = µ.
(e) Assume that 1 ∈ Im(H) and ﬁnd...

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.

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...

Let B = [ aij ] 20×17 be a matrix with real entries. Let x be in
R 17 , c be in R 20, and 0 be the vector with all zero entries.
Show that each of the following statements implies the other.
(a) Bx = 0 has only the trivial solution x = 0 n R 17, then (b)
If Bx = c has a solution for some vector c in R 20, then the
solution is unique.

Let A be an nxn matrix. Show that if Rank(A) = n, then Ax = b
has a unique solution for any nx1 matrix b.

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...

Let the vectors a and b be in
X =
Span{x1,x2,x3}.
Assume all vectors are in R^n for some positive integer n.
1. Show that 2a - b is in
X.
Let x4 be a vector in Rn that is not contained
in X.
2. Show b is a linear combination of
x1,x2,x3,x4.
Edit: I don't really know what you mean, "what does the question
repersent." This is word for word a homework problem I have for
linear algebra.

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...

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