Algebra II 189-251B Instructor: Dr. E. Goren.

Assignment 4

To be submitted by February 9, 12:00 In this assignment V and W are vector spaces of dimension n and m respectively, over a field 𝔽.

1. Let B be a basis for V and C a basis for W. Prove that for any m × n matrix A with entries in 𝔽 there exists a unique linear transformation T : V ⟶ W such that _{C}[T]_{B} = A.

2. Deduce from the theorems on determinants the following:

1. If a column is zero, the determinant is zero.

2. det(A) = det(A^{t}), where A^{t} is the transposed matrix.

3. If a row is zero, the determinant is zero.

4. Let A be a matrix in “block form”:

A = ( \begin{matrix}A_{1} & 0 & & & & ⋆ \\ 0 & A_{2} \\ & & . & . & . \\ 0 & & & 0 & & A_{k} \end{matrix} ) .

Here each A_{i} is a square matrix say of size r_{i}, and A_{2} starts at the r_{1} + 1 column and r_{1} + 1 row, etc. Prove that

det(A) = det(A_{1}) det(A_{2}) ⋯ det(A_{k}).

Conclude that the determinant of a triangular matrix is given by

det ( \begin{matrix}a_{11} & & & & & ⋆ \\ 0 & a_{22} \\ & & . & . & . \\ 0 & & & 0 & & a_{kk} \end{matrix} ) = a_{11}a_{22} ⋯ a_{kk}.

(Here each a_{ii} is a scalar).

3. Calculate the following series of determinants.

1. det ( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) , det ( \begin{matrix}0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{matrix} ) , det ( \begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix} ) , ⋯

2. det ( \begin{matrix}1 & 1 \\ 1 & 1 \end{matrix} ) , det ( \begin{matrix}1 & 1 & 0 \\ 1 & 1 & 1 \\ 0 & 1 & 1 \end{matrix} ) , det ( \begin{matrix}1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 \\ 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 \end{matrix} ) , det ( \begin{matrix}1 & 1 & 0 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 \end{matrix} ) , ⋯

3. det ( \begin{matrix}x & −a_{2} \\ 1 & x+a_{1} \end{matrix} ) , det ( \begin{matrix}x & 0 & a_{3} \\ 1 & x & −a_{2} \\ 0 & 1 & x+a_{1} \end{matrix} ) , det ( \begin{matrix}x & 0 & 0 & −a_{4} \\ 1 & x & 0 & a_{3} \\ 0 & 1 & x & −a_{2} \\ 0 & 0 & 1 & x+a_{1} \end{matrix} ) , ⋯

4. Prove the following formula (the Vandermonde determinant):

det (\begin{matrix}1 & x_{1} & x_{1}^{2} & ⋯ x_{1}^{n−1} \\ ⋮ & & ⋮ & & ⋮ \\ 1 & x_{n} & x_{n}^{2} & ⋯ x_{n}^{n−1} \end{matrix}) = ∏_{i>j}(x_{i} − x_{j})

For example, for n = 2, 3 we have

det ( \begin{matrix}1 & x_{1} \\ 1 & x_{2} \end{matrix} ) = (x_{2} − x_{1}), det ( \begin{matrix}1 & x_{1} & x_{1}^{2} \\ 1 & x_{2} & x_{2}^{2} \\ 1 & x_{3} & x_{3}^{2} \end{matrix} ) = (x_{2} − x_{1})(x_{3} − x_{1})(x_{3} − x_{2}).