Linear independence

Japanese school year: University year 1

What you learn

This section covers the precise definition and criteria for linear independence and linear dependence among a set of vectors. This concept is fundamental for defining bases and dimension in vector spaces without redundancy. Prior knowledge of vector linear combinations and techniques for solving linear systems is essential.

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Key points

This defines linear independence for a set of vectors. It means the only way to scale and add vectors $\boldsymbol{v}_1, \dots, \boldsymbol{v}_k$ to yield the zero vector $\boldsymbol{0}$ is for all scalar coefficients $c_1, \dots, c_k$ to be 0.

c_1 \boldsymbol{v}_1 + \dots + c_k \boldsymbol{v}_k = \boldsymbol{0} \implies c_1 = \dots = c_k = 0

Use this determinant test to check if $n$ vectors in an $n$ -dimensional space are linearly independent. Form a square matrix with them; if the determinant is non-zero, they are independent.

\det([\boldsymbol{v}_1 \ \dots \ \boldsymbol{v}_n]) \neq 0

This formula uses matrix rank to check if a set of vectors is linearly independent. When the rank of the matrix formed by lining up these $k$ vectors equals $k$ , they are linearly independent.

\operatorname{rank}([\boldsymbol{v}_1 \ \dots \ \boldsymbol{v}_k]) = k

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Practice 1Coming soon