quiz 7 to study for Test 2

rank of matrix =

number of pivot columns in RREF of A

The null space N(A) is a subspace of R^k, k =

number of COLUMNS in A also equal to row space C(A^T)

The column space C(A) is a subspace of R^j j =

number of ROWS in A also equal to left null space N(A^T)

The row space C(A^T) is a subspace of Rⁱ i =

number of COLUMNS in A also equal to null space N(A)

The left null space N(A^T) is a subspace of R^s s =

number of ROWS in A also equal to column space C(A)

The dimension of the null space N(A) =

(number of columns in A) - (rank of A) or (n - r)

The dimension of the column space C(A) =

rank of A also equal to row space C(A^T)

The dimension of the row space C(A^T) =

rank of A also equal to column space C(A)

The dimension of the left null space N(A^T) =

(number of rows of A) - (rank of A) or (m - r)

basis for the column space C(A) =

the pivot columns values in A, NOT their values in the RREF of A

basis for the row space C(A^T) =

the non zero rows each as vectors from the RREF of A

basis for the null space N(A) (hint: recall that the set of special solutions to Ax = 0 is a basis for N(A)) =

the free variables’ vectors (the vectors you find basically when you look for the span of N(A))

basis for the left null space N(A^T) =

A^Ty = 0, you take the reduced echelon of A^T and set each row equal to zero then solve for the free variable(s)’ vector, then you have your basis

p vector =

x * a

p = proj_ab =

(a^Tb/a^Ta) * a

Projection Matrix P =

a^Ta/a^Ta or (a * a^T)/a^Ta

If b is in C(A), then there…

is a solution of Ax = b

If b is NOT in C(A), then we can…

project b onto a vector p in C(A) and solve Ax = p for the approximate solution

error e =

b - p

approximate x vector =

(A^TA)^-1A^Tb

projection matrix P =

A(A^TA)^-1A^T

[a b]^-1
[c d] =

1/(ad - bc)[d -b]
[-c a]