Linear Algebra

Find a basis for the null space of the given matrices.

• Step 1: Row reduce to row reduced echelon form (RREF)

• Step 2: Set Ax = 0

• Step 3: basis is the columns of the free vars

Find a basis for the column space of the given matrices.

• Step 1: Row reduce to echelon form

• Step 2: Find pivot columns from row reduced matrix

• Step 3: Extract pivot columns from original matrix

Verify if a vector v is in the column space of the matrix A.

• Step 1: Row reduce to echelon form

• Step 2: Number of pivots = number of dimensions

Find a basis for the space H spanned by the vectors.

• Step 1: Row reduce to echelon form

• Step 2: Find pivot columns from row reduced matrix

• Step 3: Extract pivot columns from original matrix

Verify if a vector is in the null space of the matrix.

• Step 1: Multiply A * v (in that order) 

  • If the entire vector is 0 (Av = 0), the vector v IS in the null space of A

  • If the entire vector is NOT 0, the vector v IS NOT in the null space of A