Winter 2015
MATH 504: Abstract linear algebra
Branko Ćurgus

Thursday, March 12, 2015

4 distinct eigenvalues with lengths of Jordan chains
       (4,3,2,1,1), (5, 2, 2), (7), (1, 1, 1)

3 distinct eigenvalues with lengths of Jordan chains
       (6, 4), (3, 3, 3), (8, 2, 1)

6 distinct eigenvalues with lengths of Jordan chains
       (4, 2), (2, 2, 1), (2, 1, 1), (3, 2, 2), (3, 1), (2, 2)

3 distinct eigenvalues with lengths of Jordan chains
       (15), (8), (7)

Thursday, March 5, 2015

Friday, February 27, 2015

Thursday, February 26, 2015

Friday, February 20, 2015

Wednesday, February 18, 2015

Tuesday, February 10, 2015

Thursday, January 29, 2015

Tuesday, January 27, 2015

Monday, January 26, 2015

  • The next three problems for Assignment 1 are as follows:
    • Problem 4. Let $D$ be a finite set and let $\mathbb F$ be a scalar field. Then the set of all functions defined on $D$ with values in $\mathbb F$ is a vector space over $\mathbb F$ with the addition and scalar multiplication of functions defined pointwise. This space is denoted by ${\mathbb F}^D$.
      • (a) Prove that ${\mathbb F}^D$ is finite dimensional if and only if $D$ is finite.
        (b) If $D$ is finite, then $\dim \bigl({\mathbb F}^D\bigr) = |D|$.
    • Problem 5. Let $\mathcal V$ be a vector space over $\mathbb F$. Let $\mathcal A$ be a linearly independent subset of $\mathcal V$. Let $u \in \mathcal V$ be arbitrary. By $u + \mathcal A$ we denote the set of vectors $\{u+v : v \in \mathcal A \}$.
      • (a) Prove the following implication. If $w \notin {\rm span}\, \mathcal A$, then $ w + \mathcal A$ is a linearly independent set.
        (b) Is the converse of the implication in (a) true?
        (c) Let $\alpha_1,\cdots,\alpha_n \in \mathbb F$, let $v_1,\ldots, v_n$ be distinct vectors in $\mathcal A$ and let $w = \alpha_1 v_1 + \cdots + \alpha_n v_n$. Find a necessary and sufficient condition (in terms of $\alpha_1, \ldots, \alpha_n$) for the linear independence of the vectors $v_1 + w, \ldots, v_n + w$.
    • Problem 6. Let $\mathcal V$ be a vector space over $\mathbb F$ and $T \in {\mathcal L}(\mathcal V)$. Assume that there exists a function $f: \mathcal V \rightarrow \mathbb F$ such that $Tv = f(v)\, v$ for each $v \in \mathcal V$. Prove that $T$ is a multiple of the identity mapping. That is, there exists $\alpha \in \mathbb F$ such that $Tv = \alpha\, v$ for each $v \in \mathcal V$. (A plain English explanation: The equation $Tv = f(v)\, v$ is telling us that $T$ scales each vector in $\mathcal V$ by the scaling coefficient $f(v)$. The point of the problem is to prove that $T$ must scale each vector by the same coefficient. This is a consequence of the linearity of $T$.)

  • Saturday, January 24, 2015

    Thursday, January 22, 2015

    Monday, January 19, 2015

    Tuesday, January 13, 2015

    Monday, January 12, 2015

    Thursday, January 6, 2015