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Vectors

The set of vectors {[ 1 -1] , [ 1 -1] , [ 2 -1] }
[ 2 0 ] [ -1 0] [ -1 0]

from M_2(R) is:
A. linearly dependent
B. linearly independent
C. orthogonal
D. a spanning set for M_2(R)
E. a basis for M_2(R)

Solution Preview

I presume that each vector in this set is a 2*2 matrix. Then we have:

{[ 1 -1] , [ 1 -1] , [ 2 -1] }
[ 2 0 ] [ -1 0] [ -1 0]

part a and b.

Consider

[1 -1]
A1 :=
[2 0]

[1 -1]
A2 :=
[-1 0]

[2 -1]
A3 :=
[-1 0]

Then we define mA1+nA2+pA3=0 (zero matrix) and solve for m, n and p:

...

Solution Summary

This shows how to determine the type of vectors from a matrix.

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