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 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.
Part 2: Determinant and Matrix Properties
Calculating the determinant of matrix A involves applying expansion or properties of determinants. If the determinant is non-zero, A is invertible and its columns are linearly independent. Conversely, a zero determinant indicates that A is singular and the columns are linearly dependent. These properties are crucial in solving systems of equations and understanding the invertibility of matrices.
Part 3: Invertibility and Column Space Dimension
For matrices ■ and ■′ , invertibility is assessed via the determinant or rank. The Uniqueness Theorem states that a square matrix is invertible iff its determinant is non-zero. The Rank Theorem relates the rank of the matrix to the dimension of its column space, informing about the linear independence of columns. Solving systems like (■ ■′)■′ = ■ without direct matrix multiplication can often be approached using properties of invertible matrices and their inverses or pseudo-inverses.
Conclusion
Analyzing vectors in subspaces, understanding matrix invertibility, and determining the dimensions of column spaces are interconnected areas in linear algebra. They underpin the solutions of linear systems, the structure of vector spaces, and the properties of transformation matrices. Mastery of these concepts enables solving complex problems efficiently and accurately.
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