Vector Spaces

Basis Check

Determine whether a set of k vectors forms a basis for R^n. The vectors form a basis iff they are linearly independent (rank = k) AND k = n. When rank < k the vectors are dependent and cannot span R^n.

Example

Reduce and combine count with independence.

highlighted = computed this step

Step 1 — Set up reduction

Set up the augmented matrix for row reduction.

R=[1224]R=\begin{bmatrix}1&2\\2&4\end{bmatrix}

Step 2 — Update row 2

Row-reduce with R2 <- R2 - 2*R1.

R=[1200]R=\begin{bmatrix}1&2\\\hl{0}&\hl{0}\end{bmatrix}

Step 3 — Compare rank and count

Count the pivot columns: rank = 1.

rank⁡=1\operatorname{rank}=\hl{1}

Step 4 — State verdict

These vectors do NOT form a basis of R^2 (rank 1, so a free column exists).

not a basis of R2\text{not a basis of }\mathbb{R}^{2}
basis A basis for R^n is a set of n linearly independent vectors that spans R^n. Given k candidate vectors, row-reduce the matrix A = [v1 | … | vk]: if the rank equals n and k equals n, the set is a basis.