Vector Spaces

Span Check

Determine whether a vector b lies in the span of {v1, v2} by forming the augmented matrix [v1 | v2 | b] and row-reducing. If the system is consistent, b is in the span; otherwise it is not.

span The span of a set of vectors {v1, v2, …} is the set of all linear combinations c1·v1 + c2·v2 + …. A vector b is in the span iff the augmented system has at least one solution.

Example

Reduce the augmented system and state consistency.

highlighted = computed this step

Step 1 — Set up reduction

Set up the augmented matrix for row reduction.

R=[1273410]R=\left[\begin{array}{cc|c}1&2&7\\3&4&10\end{array}\right]

Step 2 — Update row 2

Row-reduce with R2 <- R2 - 3*R1.

R=[1270-2-11]R=\left[\begin{array}{cc|c}1&2&7\\\hl{0}&\hl{-2}&\hl{-11}\end{array}\right]

Step 3 — Check consistency

Check consistency: no contradiction row appears.

consistent\hl{\text{consistent}}

Step 4 — State verdict

Yes — b is in the span (the system is consistent).

in span\hl{\text{in span}}