LINEAR ALGEBRA EXAM 2 Q&A

LINEAR ALGEBRA EXAM 2 Q&A
Getting x using Coordinate matrix and x-coordinate vector - Answer-x = Pb*[x]b

Getting x-coordinate vector using Inverse of coordinate matrix and x-vector - Answer-
[x]b = Pb^-1*x

What is dim(Nul A)? - Answer-Number of free variables, n - rank A.

What is dim(Col A)? - Answer-Number of pivots

What is dim(Row A)? - Answer-Number of pivots

What is the rank of a matrix? - Answer-The number of pivots matrix has in REF form

What is isomorphism? - Answer-A linear transformation that is both one-to-one and onto

The coordinate vector x? [x]B - Answer-[x]B = [c1 c2 ... cn] <=> x = c1b1 + c2b2 + ... +
cnbn

How to check if v is in Nul A? - Answer-A*v = 0 WE NEED TO PERFORM MATRIX
MULTIPLICATION

How to check if v is in Col A? - Answer-Ax = v WE NEED TO SOLVE LINEAR SYSTEM
Ax = v.

Rank-Nullity Theorem - Answer-rank(A) + nullity(A) = number of columns in A

Nullity(A) means the dimension of Null Space of A

Dimension of Row(A) and Row(A^T) - Answer-dim(Row(A)) = dim(Row(A^T)) -> always
true

Dimension of vector space and size of matrix for Mnxm - Answer-m*n

dim(Nul(A^T) = - Answer-dim(Nul(A^T)) = # cols of A^T - rank(A)