[Intro] What is a vector space? Define it properly! Eight axioms, all of them, recite them back to me! Additive identity, where is it? Show your work! Associativity holds? Then prove it! [Verse 1] Is this a vector space? Prove it, every axiom! Closure under addition, verify the sum! Scalar multiplication, check it for every case! Is the zero vector present? Find it, state the place! You call that a basis? Are those vectors independent? Reduce them, check the rank! Span the whole space, confirm the dimension is right! Count the basis vectors, does the count match the dimension? [Pre-Chorus] Write the definition, state the conditions! Apply the theorem, verify each hypothesis! [Chorus] Row reduce that matrix, now! Echelon form, I will show you how! Find the null space, find the rank! Pivot columns, mark them clear! State the dimension of each subspace here! [Verse 2] What is the determinant? Expand along the row! Cofactor expansion, every term, compute them all below! If det is zero, the matrix is singular! The columns are dependent, state that in particular! Solve det of A minus lambda I equals zero! Characteristic polynomial, factor it! For each eigenvalue, solve the homogeneous system! Find the eigenspace, write the basis, do not miss them! [Pre-Chorus] Write the definition, state the conditions! Apply the theorem, verify each hypothesis! [Chorus] Row reduce that matrix, now! Echelon form, I will show you how! Find the null space, find the rank! Pivot columns, mark them clear! State the dimension of each subspace here! [Bridge] A is diagonalizable if and only if the sum of the eigenspace dimensions equals n. Write the change of basis matrix P. Columns are eigenvectors, ordered by eigenvalue. P inverse times A times P equals diagonal. Verify it. Compute it. Show every step. [Verse 3] Apply Gram Schmidt to your linearly independent set! Project onto the previous vector, then subtract the component! Normalize each result, confirm the dot product is zero! Orthonormal basis, state each vector and its norm! Least squares, multiply both sides by A transpose! Solve the normal equations, A transpose A times x equals A transpose b! Find the orthogonal complement of the column space! Every vector in the null space of A transpose, state its place! [Final Chorus] Row reduce that matrix, now! Echelon form, I will show you how! Find the null space, find the rank! Pivot columns, mark them clear! State the dimension of each subspace here! [Outro] State the rank nullity theorem, every term! Dimension of the null space plus rank equals n, confirm! Give me a basis for the column space, write it down! Verify with row reduction, every pivot, every column found!
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