Linear Independence and Linear Dependence Quiz

Linear Independence of Vectors

• A set of vectors is linearly independent if none of the vectors in the set can be written as a linear combination of the others.
• If a set of vectors is linearly independent, there is only one solution to the vector equation.
• Linear independence implies that the constants in the vector equation are all zero, except for one.
• A multiple of one vector does not affect linear independence.

Linear Dependence of Vectors

• A set of vectors is linearly dependent if one of the vectors in the set can be written as a linear combination of the others.
• Linearly dependent vectors imply that there are infinitely many solutions to the vector equation or no solution at all.
• The characteristic of linearly dependent vectors is that one vector is a linear combination of the others.

Determining Linear Independence

• To determine if vectors are linearly independent, inspect the matrix formed by the vectors as columns.
• A set of vectors is considered linearly dependent if one column is a linear combination of the others.
• A column in the matrix is considered a pivot column if it is not a linear combination of the previous columns.

Pivot Columns and Linear Independence

• If a column in the matrix is not a pivot column, it means it can be written as a linear combination of the previous columns, implying linear dependence.
• If a column in the matrix is a pivot column, it means it is not a linear combination of the previous columns, implying linear independence.
• A set of vectors is not linearly independent if one column is not a pivot column.