Theorems Concepts - LA

Question 1

Q: what is Ax=b ?

Answer 1

Q: what is Ax=b ?

output Ax i.e b, is a linear combination of columns of A.
Every matrix transforms its row space onto its column space.

Question 2

Q: when does solution exists? for Matrix form - Ax=b

Answer 2

Q: when does solution exists? for Matrix form - Ax=b

[Solution exists for possible b’s only if all column vectors are linearly independent, also vectors span throught space]

Question 3

Q: Fundamental Theorem of Linear Algebra, Part I [fundamental spaces & its dimension]

Answer 3

Q: Fundamental Theorem of Linear Algebra, Part I [fundamental spaces & its dimension]

1. C (A) = column space of A; dimension r.
2. N (A) = nullspace of A; dimension n − r.
3. C (A^T ) = row space of A; dimension r.
4. N (A^T ) = left nullspace of A; dimension m − r.
   Picture of 4 Blocks of C(A), C(A^t), N(A) and N(A^t)
   A(xp+xn)=b

Question 4

Q: What are different cases for Ax=b?

Answer 4

Q: What are different cases for Ax=b?

Solution of Ax=b [All 4 cases]
Case 1 - A[mxn], full column rank, rank r=n : Free variables= n-r=0, unique solution
Case 2 - A[mxn], full row rank, rank r=m : Free variable= n-r=n-m, always a solution
Sub-Case 3 - A[mxn], rank r=m=n : unique solution
Case 4 - A[mxn], rank r

Question 5

Q: 3D Fundamental Theorem of Linear Algebra, Part II [othogonal complements]

Answer 5

Q: 3D Fundamental Theorem of Linear Algebra, Part II [othogonal complements]

Nullspace and Row space are orthogonal complements in Rn, from Ax=b.
Similarly, N(A^T) and C(A) are orthogonal complements in Rm.

Question 6

Q: Every m by n matrix of rank r reduces to (m by r) times (r by n): A- matrix, R - its row reduced echelon form

Answer 6

Q: Every m by n matrix of rank r reduces to (m by r) times (r by n): A- matrix, R - its row reduced echelon form
A = (pivot columns of A)(first r rows of R)) = (COL)(ROW)