Find bases of the domain and codomain that place the...

Find bases of the domain and codomain that place the following matrices in the canonical form (7.33). Use (7.32) to check your answer. (a) $\left(\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right)$, (b) $\left(\begin{array}{rrr}1 & -3 & 4 \\ -2 & 6 & -8\end{array}\right),(c)\left(\begin{array}{rr}2 & 3 \\ 0 & 4 \\ -1 & 1\end{array}\right)$, (d) $\left(\begin{array}{rrr}1 & 2 & 1 \\ 1 & -1 & -1 \\ 2 & 1 & 0\end{array}\right)$, (e) $\left(\begin{array}{rrrr}1 & 3 & 0 & 1 \\ 2 & 6 & 1 & -2 \\ -1 & -3 & -1 & 3 \\ 0 & 0 & -1 & 4\end{array}\right)$.

Find bases of the domain and codomain that place the following matrices in the canonical form (7.33). Use (7.32) to check your answer.
(a) $\left(\begin{array}{ll}1 & 2 \\ 2 & 1\end{array}\right)$,
(b) $\left(\begin{array}{rrr}1 & -3 & 4 \\ -2 & 6 & -8\end{array}\right),(c)\left(\begin{array}{rr}2 & 3 \\ 0 & 4 \\ -1 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rrr}1 & 2 & 1 \\ 1 & -1 & -1 \\ 2 & 1 & 0\end{array}\right)$,
(e) $\left(\begin{array}{rrrr}1 & 3 & 0 & 1 \\ 2 & 6 & 1 & -2 \\ -1 & -3 & -1 & 3 \\ 0 & 0 & -1 & 4\end{array}\right)$.

Key Concepts

Basis

A basis is a set of linearly independent vectors that spans a vector space. In the context of linear maps, choosing bases for both the domain and the codomain allows us to represent the transformation as a matrix, making it easier to analyze and simplify the map by changing coordinates.

Change of Basis

Changing the basis of a vector space involves selecting a new set of basis vectors, which can simplify the representation of a linear transformation. This process is executed using invertible matrices and is essential for converting a matrix into its canonical form, revealing inherent structural properties.

Canonical Form

A canonical form of a matrix is a simplified version, such as a block diagonal or structured form, that uniquely represents the linear transformation up to the choice of basis. It typically highlights key features like rank, nullity, and invariant subspaces, facilitating comparisons and further analysis.

Rank-Nullity Theorem

The Rank-Nullity Theorem states that for a linear transformation, the sum of its rank (the dimension of the image) and nullity (the dimension of the kernel) equals the dimension of the domain. This theorem is crucial when selecting bases that bring the matrix to a form where the nonzero blocks correspond to the rank and the zero blocks to the nullity.

Matrix Equivalence

Matrix equivalence refers to the idea that two matrices represent the same linear transformation under different choices of bases. By finding appropriate bases for the domain and codomain, a given matrix can be transformed into a canonical form, making it easier to compare and understand its structure.

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