Question

For each of (a)-(d) below, decide whether the statement is True or False. If True, explain why. If False, give a counterexample.

(a) Two non-parallel planes in R 3 will always meet in a line.

(b) Let v, x1, x2 ∈ R 3 with x1 6= x2. Define two lines L1 and L2 by

L1 : (x, y, z) = x1 + sv, s ∈ R,

L2 : (x, y, z) = x2 + tv, t ∈ R.

Then the lines L1 and L2 will never intersect.

(c) Let v1, v2, v3 be three non-zero vectors in R 3 . Then 0 (the zero vector) is a linear combination of v1, v2 and v3. (d) If A and B are two 2 × 2 matrices, and AB = 02×2 (where 02×2 is the 2 × 2 zero matrix), then at least one of A or B must be the zero matrix.

Answer 1

a)

Yes two parallel planes always meat in a line. And it can be shown by linear algebra since if you consider the following system of the two plane equations

• ax+by+cz=d
• a′x+b′y+c′z=d′a′

with a,b,c not multiple of a',b',c' the rank of the associated matrix is 2 and we always have infinitely many solutions.

C)

It is always true.

FUNDAMENTAL THEOREM IN THREE-DIMENSIONS:

Let a , b and c be three non-zero non-coplanar vectors in space. Then any vector r in space can be expressed uniquely as a linear combination of a , b and c i.e there exist unique l, m, n ∈ R such that la + mb + n c = r.

Note: if we take coplaner vectors then one can be written as a linear combination of others and there combination can always make zero.

d)

It is not true

Consider the matrices A and B

Sorry I am not understanding problem (b) what is the mean of x1 6 = x2