Basis, Rank, and Transformation Workshop

Retrieval Prompts

1. State the definition of linear independence.
2. State the definition of basis.
3. State the rank-nullity relationship.
4. State the test for linearity.
5. State how matrix columns relate to images of basis vectors.

Compare and Distinguish

Separate these pairs clearly:

• spanning set versus basis
• independent set versus maximal independent set
• transformation versus matrix representation
• invertible map versus merely injective map

Common Mistake Check

Find the error in each statement:

1. "Three vectors in R^3 always form a basis."
2. "If two matrices differ, they must represent different transformations."
3. "A linear map can include a constant offset as long as the matrix part is linear."
4. "Dimension equals the number of equations used to describe the space."

Mini Application

Choose a linear map from R^2 to R^2, such as a shear, reflection, or scaling-plus-shear. Then:

1. write its matrix in the standard basis
2. apply it to two basis vectors and verify the columns
3. decide whether it is invertible
4. describe its column space and nullspace
5. explain how a change of basis could make the same action easier or harder to interpret

Evidence Check

This page is complete only if you can move fluently among these four descriptions of the same object:

• equations
• matrix
• subspaces
• transformation language