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2024/2025MATH 218 TEST 2 Questions And
Answers Latest Update
what factorization forms a basis for left null space ANS✔✔ EA=R
if A is an mxn matrix with rank r (easily seen wtih R), the last m-r rows of E form basis of left null
space.
how to verify if vector is in column space ANS✔✔ row reduce A augmented with b
if there's no pivot in the augmented column, b belongs to the column space
how to verify if a vector is in the null space ANS✔✔ multiply the matrix by the vector
if the answer is 0 it is in the null space
how to verify vector is in the row space ANS✔✔ augment with the vector and row reduce
if there is no pivot in the augmented column the vector belongs to the row space
how to verify if vector is in the left null space ANS✔✔ check if the transpose of A times the vector
is 0
or
check if the transpose of the vector times A is 0
how to produce vectors in the null space ANS✔✔ augment A with the 0 vector and parametrize.
, 2024/2025
null space is the vectors multiplied by the scalars
same as saying null space is spanned by those vectorsmu
how to produce vectors in the column space ANS✔✔ multiply matrix by any vector x and resulting
matrix is in the column space
how to produce vectors in the row space ANS✔✔ multiply transpose of matrix by any vector x and
resulting matrix is in the row space
basis of null space ANS✔✔ same process as producing vectors in the null space
basis is that span
basis of left null space ANS✔✔ same process as producing vectors in the left null space
basis is that span
basis of column space ANS✔✔ the pivot columns of A
note: though the pivot columns of rref(A) match up, NOT the same entries, so you must have the
matrix A
OR
nonzero row of the rref of the matrix transposed
basis of row space ANS✔✔ the pivot columns of the transpose of A
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