MATH 136 - Midterm

MATH 136 - Midterm
Linearly independent set, subset of a subspace and vice versa - ANS-Basis

Contains zero vector, closed under scalar multiplication and addition - ANS-Subspace

When the products of two vectors equal to zero - ANS-Orthgonal

Has all of the coefficients of the linear system. [A] - ANS-Coefficient Matrix

Contains the coefficient matrix and the vector. [A | b] - ANS-Augmented Matrix

Multiplying a row by a non-zero scalar (cRi) - ANS-Elementary Row Operation 1

Adding a multiple of one row to another (Ri+cRj) - ANS-Elementary Row Operation 2

Switching two rows (Ri ↔ Rj) - ANS-Elementary Row Operation 3

First non-zero entry is each non-zero row. - ANS-Leading One

The number of leading ones in an RREF Matrix. rank ≤ min(m,n) - ANS-Rank (2.2.4)

(1) All rows with zero are at the bottom
(2) The first non-zero entry of each row is a leading one
(3) The leading one is the only non-zero entry in its column
(4) Each leading one is to the right of the leading one in the row above - ANS-Reduced Row
Echelon Form

If a j-th column of RREF does not contain a leading one, then xj is a free variable. - ANS-Free
Variable

(1) rank(A) is less than rank([A | b]) iff the system is inconsistent
(2) If system [A | b] is consistent, then the system contains (n - rank A) free variables
(3) The system [A | b] is consistent ∀b∈R^m iff rank A = m - ANS-System-Rank Theorem
(2.2.5)

Let [A | b] be a consistent system of m linear equations in n variables with RREF [R | c]. If rank
A = k < n, then a vector equation of the solution set of [A | b] has the form:

x = d + t1v1 + ... + tn-kvn-k, t∈R