The Vector ■ Is In A Subspace H With A Basis Quiz 5: (Each 5 points) 1) The vector ■ is in a subspace H with a basis vector ■ = {■■, ■■} where ■■ = [ ■■ ■■ ] and ■■ = [ -■■ -■■ ], ■ = [ -■■ -■■ ■■ ]. Find the B co-ordinate vector of ■. By inspection write another different basis ■′ of H and find the ■′ co-ordinate vector of the same ■. 2) Find the DETERMINANT of the following matrix : A = [ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■■ ■■ ■■ ■■ ■■ ]. Answer the followings. Explain: (i) Is A invertible? (ii) Are the columns of A linearly Independent? 3) Let A = [ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ■ ] and B = [ -■ -■ -■ -■ -■ -■ -■ -■ -■ -■ -■ -■ -■ -■ -■ -■ -■ -■ -■ -■ -■ -■ -■ -■ -■ ]. ANSWER AND EXPLAIN THE FOLLOWING: (i) Is the matrix ■ of ■ derived from ■ © invertible? (Do not compute AB) (ii) Solve (■ ■ ©)■' = ■ without computing the matrix AB or its transpose. (iii) What is the dimension of the column space of (■ ©■ )■'? (Hint: Use Determinant Invertible Matrix Theorem and Rank Theorem)
Paper For Above instruction Understanding subspaces and bases is fundamental in linear algebra, especially when determining the coordinate vectors of vectors relative to a basis, exploring matrix properties such as invertibility, and analyzing the dimensions of column spaces. This paper discusses the procedures involved in these topics, elucidating through examples and theoretical explanations. Part 1: Coordinates of a Vector in a Subspace with a Basis Given a vector ■ in a subspace H with a basis ■ = {■■, ■■}, where ■■ and ■■ are vectors in ■², the goal is to find ■-coordinates of ■. Suppose ■■ = [■■, ■■] and ■■ = [-■■, -■■], while ■ = [-■■, -■■, ■■]. To determine the coordinate vector, we express ■ as a linear combination of the basis vectors: ■ = α■■ + β■■ which leads to solving the system: [■■ ■■] [α β]■ = ■ By inspecting the basis vectors, a different basis ■′ can be constructed by selecting vectors that span H but differ from ■, for example, a basis involving linear combinations of ■■ and ■■. The coordinate vector of ■ relative to ■′ can then be similarly computed.