Question

Prove or disprove: If the columns of B(n×p) ? R are linearly independent as well as those of A, then so are the columns of AB (for A(m×n) ? R ).

Answer #1

_{1},b_{2},…,b_{p}], where
b_{i} (1 ? i ? p) is the ith column vector of
B. Further, AB = [Ab_{1},Ab_{2},…,Ab_{p}] .
Let us assume that the columns of a mx n matrix A and a nxp matrix
B with real entries are linearly independent while the columns of
AB are linearly dependent. Then there exist real scalars
x_{i} (1 ? i ? p), not all zero such that
x_{1}Ab_{1}+x_{2} Ab_{2}
+…+x_{p} Ab_{p} = 0. Then
A(x_{1}b_{1}
+x_{2}b_{2}+…+x_{p}b_{p}) = 0 so
that x_{1}b_{1}
+x_{2}b_{2}+…+x_{p}b_{p} = 0.
However, since the columns of B are linearly independent, this
means that all the x_{i} s are 0 which is a contradiction.
Hence the columns of AB are also linearly independent.

(§2.1) Let a,b,p,n ∈Z with n > 1.
(a) Prove or disprove: If ab ≡ 0 (mod n), then a ≡ 0 (mod n) or
b ≡ 0 (mod n).
(b) Prove or disprove: Suppose p is a positive prime. If ab ≡ 0
(mod p), then a ≡ 0 (mod p) or b ≡ 0 (mod p).

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.

Prove or disprove that [(p → q) ∧ (p → r)] and [p→ (q ∧ r)] are
logically equivalent.

Prove or disprove that if (xn) is an unbounded sequence in R,
then there exists n0 belongs to N so that xn is greater than 10^7
for all n greater than or equal to n0

Prove or disprove that for any events A and B,
P(A) + P(B) − 1 ≤ P(A ∩ B) ≤ min{P(A), P(B)}.

Let x1, x2, ..., xk be linearly independent vectors in R n and
let A be a nonsingular n × n matrix. Define yi = Axi for i = 1, 2,
..., k. Show that y1, y2, ..., yk are linearly independent.

If A and B are matrices and the columns of AB are independent,
show that the columns of B are independent.

Prove that any set of vectors in R^n that contains the vector
zero is linearly dependent.

(a) Prove or disprove the statement (where n is an integer): If
3n + 2 is even, then n is even.
(b) Prove or disprove the statement: For irrational numbers x
and y, the product xy is irrational.

. Let f : Z → N be function.
a. Prove or disprove: f is not strictly increasing. b. Prove
or disprove: f is not strictly decreasing.

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 18 minutes ago

asked 19 minutes ago

asked 26 minutes ago

asked 27 minutes ago

asked 35 minutes ago

asked 35 minutes ago

asked 35 minutes ago

asked 54 minutes ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago

asked 1 hour ago