Linear Algebra: Row Space, Null Space, Determinants, and Gram-Schmidt

Finding the Basis of a Row Space

The easiest way to find the basis of a row space is to reduce matrix A to Reduced Row Echelon Form (RREF). The nonzero row vectors of R (which contain the leading 1s, or pivots) form a basis for row(A).

Finding the Basis of the Kernel

The following four steps outline the most effective method for finding a basis for null(A):

1. Reduce A to RREF (R): Find the Reduced Row Echelon Form (R) of the matrix A.
2. Solve the Homogeneous System: Use the RREF, R, to solve the equivalent homogeneous system Rx=0.
3. Identify and Parameterize Variables:
   • Identify the leading variables (those corresponding to columns containing a leading 1 or pivot in the RREF) and the free variables.
   • Solve for the leading variables in terms of the free variables.
   • Set each free variable equal to a parameter (e.g., s, t).
4. Express the Solution as a Linear Combination: Substitute the parameterized expressions back into the solution vector x and write the result as a linear combination of vectors. The vectors resulting from this linear combination form a basis for null(A).

Understanding Row Space and Null Space

• Row Space: The row space of A, denoted row(A), is the subspace spanned by the rows of A.
• Null Space: The null space of A, denoted null(A), is the subspace of Rⁿ consisting of solutions x to the homogeneous linear system Ax=0.

Condition for Orthogonal Complement

• A vector x is in the orthogonal complement of row(A), denoted (row(A))^⊥, if and only if x is orthogonal to every row of A.
• The condition for orthogonality between two vectors u and v in Rⁿ is u·v=0. Using the matrix definition, the dot product x·y is equivalent to the matrix product x^Ty.

Connecting the Concepts

Let A be an m×n matrix. The vector Ax is zero (x∈null(A)) if and only if x is orthogonal to every row of A. Since the row space is spanned by the rows of A, if x is orthogonal to every row, it is necessarily orthogonal to every linear combination of those rows. This confirms that (row(A))^⊥ = null(A).

Computing Determinants

• 2×2 Matrix: For A=(a₁₁, a₁₂, a₂₁, a₂₂), the determinant is a₁₁a₂₂ - a₁₂a₂₁.
• 3×3 Matrix (Sarrus' Rule): A method where the first two columns are copied to the right to sum products along diagonals.
• Row Reduction Method: The most efficient way to compute determinants for large matrices by reducing to upper triangular form.

The Gram-Schmidt Process in R³

The Gram-Schmidt process constructs an orthogonal basis O={v₁,v₂,v₃} and then an orthonormal basis Q={q₁,q₂,q₃} from a given basis B={x₁,x₂,x₃}.

Step 1: Establish the first orthogonal vector

v₁ = x₁

Step 2: Calculate the second orthogonal vector

v₂ = x₂ - proj_v1(x₂)

Step 3: Calculate the third orthogonal vector

v₃ = x₃ - proj_v1(x₃) - proj_v2(x₃)

Step 4: Normalize the orthogonal vectors

To transform the orthogonal set into an orthonormal set, divide each vector v_i by its length (norm) to get q_i.