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חשבון אינפיניטסימלי 2 - אוסף נוסחאות
כלל המכפלה וכלל המנה:
The intermediate value theorem
Let \(a,b,\gamma\in\mathbb{R}\) s.t. \(a<b\) and let \(f:D\rightarrow\mathbb{R}\) be a continuous function s.t. \(\left[a,b\right]\subseteq D\) and \(f(a)\ne f(b)\).
אלגברה ליניארית 2 - סיכום
Exercises 2B, Q1
Determine all vector spaces that have exactly one basis, up to reordering, and prove that no others exist. Note that this is different from the original question.
Semester 2026B, Quiz 1
Quiz 1 in the course Infinitesimal Calculus (2), semester 2026 B.
Semester 2026A, Quiz 4
Quiz 4 in the course Infinitesimal Calculus (1), semester 2026 A.
A Problem on the Convergence of a Recursive Sequence Depending on Its Initial Value
Let \(\left(a_{n}\right)_{n=1}^{\infty}\) be a sequence s.t. \(\forall n\in\mathbb{N}\quad a_{n+1}=a_{n}\cdot(1-a_{n})\).
Weierstrass Extreme Value Theorem
Let \(f:\left[a,b\right]\rightarrow\mathbb{R}\) be a continuous function from \(\left[a,b\right]\) to \(\mathbb{R}\).
Weierstrass Boundedness Theorem
Let \(f:\left[a,b\right]\rightarrow\mathbb{R}\) be a continuous function from \(\left[a,b\right]\) to \(\mathbb{R}\).
Archimedean Property
(The Archimedean Property). \(\mathbb{N}\) is not bounded from above in \(\mathbb{R}\).
Fermat’s Theorem
Let \(f:\left(a,b\right)\rightarrow\mathbb{R}\) be a function from \(\left(a,b\right)\) to \(\mathbb{R}\) and suppose that \(x_{0}\in\left(a,b\right)\) is a point where \(f\) has a local extremum. If \(f\) is…
The Bolzano-Weierstrass Theorem
When I write monotonically increasing or decreasing, I refer to the weak sense.
Semester 2026A, Exam B, Q3
Let \(V\) be a vector space over \(\mathbb{R}\) and let \((v_{1},v_{2},v_{3})\) be a linearly independent sequence of vectors from \(V\). Determine for which values \(a\in\mathbb{R}\) the sequence…
Semester 2026A, Exam A, Q1
Let \(V,W\) be vector spaces, let \(T\colon V\to W\) be a linear map, let \(\mathcal{B},\mathcal{C}\) be bases of \(V\) and \(W\) respectively, and let \(v\in V\). Prove that…
Semester 2026A, Exam A, Q4
Let \(V\) be a vector space and let \(v_{1},\dots,v_{n},u,w\in V\) be such that the sequence \((v_{1},v_{2},\dots,v_{n},u)\) is linearly dependent, while the sequence \((v_{1},\dots,v_{n},w)\) is linearly independent.
Semester 2026A, Exam A, Q6
Let \(\mathbb{F}\) be a field and let \(V=\mathbb{F}[X]_{\le3}\).
Semester 2026A, Exam A, Q7
Let \(V\) be a vector space with \(\dim V=n\), and let \(T\colon V\to V\) be a linear map such that \(\operatorname{Im}(T)\cap\ker(T)\ne\{0_{V}\}\). Prove that \(\operatorname{rk}(T\circ T)\le n-2\).
Semester 2026A, Exam A, Q9
Are the following matrices similar?…
Semester 2026A, Exam A, Q2
Let \(D\colon M_{n\times n}(\mathbb{F})\to\mathbb{F}\) be a volume function, and let \(A\in M_{n\times n}(\mathbb{F})\).
Semester 2026A, Exam B, Q1
Let \(A,B\in M_{n\times n}(\mathbb{F})\) be such that \(AB=I_{n}\). Prove that \(A\) and \(B\) are inverses of each other.
Semester 2026A, Exam A, Q3
For each \(a\in\mathbb{R}\) we will define a system of linear equations over \(\mathbb{R}\) by \(\begin{cases} 7x+2ay+z & =a\\ 8x+3ay+z & =a\\ 10x+5ay+2z & =2a \end{cases}\)
Semester 2026A, Exam A, Q8
Let \(A,B\in M_{m\times n}(\mathbb{R})\) be matrices and let \(b\in\mathbb{R}^{m}\), such that the system of linear equations \(Ax=b\) has a unique solution. Which of the following claims are true? If a claim is true…
אלגברה ליניארית 1 - הסיכום המוכלל
האוניברסיטה העברית, אלגברה ליניארית (1) - \(80134\), סמסטר 2026 א’ (תשפ“ו א’).
Semester 2025B, Exam B, Q1
Let \(V\) be a finitely generated vector space such that \(\dim\ V=n\), and let \(\left(v_{1},\dots,v_{n}\right)\) be a sequence of vectors from \(V\). Prove that \(\left(v_{1},\dots,v_{n}\right)\) is linearly…
Result 1.45
condition for a direct sum
Semester 2026A, Exam A, Q1
Prove that a bounded sequence is converging if and only if it has exactly one partial limit.