Question

determine a linearly independent set of vectors that spans the same subspace of V as that spanned by the original set of vectors.

V = R³, {(3, 1, 5), (0, 0, 0), (1, 2, –1), (–1, 2, 3)}.

Answer 1

Answer:

For the set S1, you form the matrix

⎛⎝⎜102215−11−1⎞⎠⎟∼⎛⎝⎜100211−111⎞⎠⎟∼⎛⎝⎜100210−110⎞⎠⎟∼⎛⎝⎜100010−310⎞⎠⎟

For the set S2, you form the matrix

(−21−610−2)∼(11310−2)∼(103−20−2)∼(103101)∼(1001−31)

Since we know that elementary row operations do not change row space, we see that

span(S1)=span{(1,0,−3),(0,1,1)}=span(S2).

This methods works in general. If the non-zero rows in the rrefs are different, that means span(S1)≠span(S2).

To formulate this more clearly, we know that:

Theorem. Let A and B be two m×n matrices over the same field. The following conditions are equivalent:

The matrices A and B are row equivalent.

The matrices A and B have the same row space.

The matrices A and B have the same reduced row echelon