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).

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.

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}}