numerical linear algebra

Quiz 5: (Each 5 points)

1) The vector 𝒙 is in a subspace H with a basis vector 𝑩 = {𝒃𝟏, 𝒃𝟐}

where 𝒃𝟏 = [ 𝟏 𝟑

−𝟐 ], 𝒃𝟐 = [

−𝟐 −𝟓 𝟑

], 𝒙 = [ −𝟖 −𝟏𝟕 𝟗

]. Find the B co-ordinate

vector of 𝒙. By inspection write another different basis 𝑩′of H and find the 𝑩′ co-ordinate vector of the same 𝒙.

2) Find the DETERMINANT of the following matrix :

A =

[ 𝟏 𝟏 𝟐 𝟑 𝟓 𝟏 𝟐 𝟎 𝟎 𝟎 𝟑 𝟑 −𝟒 𝟎 𝟎 𝟎 −𝟒 𝟎 𝟎 𝟎 𝟐 𝟔 𝟏 𝟕 𝟓]

Answer the followings. Explain:

(i) Is A invertible? (ii) Are the columns of A linearly Independent?

3) Let A =

[ 𝟏 𝟎 𝟎 𝟎 𝟎 𝟏 𝟐 𝟎 𝟎 𝟎

𝟑 √𝟑 𝟑 𝟎 𝟎 𝟔𝟓 𝟖 𝟗𝟏 𝟏 𝟎 𝟐 𝟔 𝟏 𝟕 𝟏]

B =

[ −𝟏 𝟖 𝟓𝟓 𝟐𝟑 𝟗 𝟎 𝟓 𝟓 𝟕𝟕 𝟑 𝟎 𝟎 𝟏 𝟑𝟒 𝟗 𝟎 𝟎 𝟎 𝟐 𝟖 𝟎 𝟎 𝟎 𝟎 𝟐]

ANSWER AND EXPLAIN THE FOLLOWING:

(i) Is the matrix 𝑨𝑩 invertible? (Do not compute AB) (ii) Solve (𝑨𝑩)𝑻𝑿 = 𝟎 without computing the Matrix AB or its

transpose.

(iii) What is the dimension of column space of (𝑩𝑨)𝑻?

(Hint:Use Determinant Invertible Matrix Theorem and Rank

Theorem)