Vector Spaces

Linear Independence

Test whether a set of vectors {v1, v2} is linearly independent by forming the matrix A = [v1 | v2] and row-reducing. The vectors are independent iff the rank equals the number of vectors (no free variable).

Example

Reduce and compare rank to column count.

highlighted = computed this step

Step 1 — Set up reduction

Set up the augmented matrix for row reduction.

R=[1235]R=\begin{bmatrix}1&2\\3&5\end{bmatrix}

Step 2 — Update row 2

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

R=[120-1]R=\begin{bmatrix}1&2\\\hl{0}&\hl{-1}\end{bmatrix}

Step 3 — Count pivots

Count the pivot columns: rank = 2.

rank⁡=2\operatorname{rank}=\hl{2}

Step 4 — State verdict

The columns are linearly INDEPENDENT (a pivot in every column).

independent\hl{\text{independent}}
linear independence Vectors are linearly independent if no vector in the set can be written as a linear combination of the others. Equivalently, the only solution to c1·v1 + … + ck·vk = 0 is c1 = … = ck = 0.