MCQ 1
If \(\langle\cdot{,}\cdot\rangle,\langle\!\langle\cdot{,}\cdot\rangle\!\rangle:V\times V\to \mathbb{K}\) are two bilinear forms, then so is \(\langle\cdot{,}\cdot\rangle+\langle\!\langle\cdot{,}\cdot\rangle\!\rangle.\)
- True
- False
MCQ 2
If \(\langle\cdot{,}\cdot\rangle,\langle\!\langle\cdot{,}\cdot\rangle\!\rangle:V\times V\to \mathbb{K}\) are two non-degenerate bilinear forms, then so is \(\langle\cdot{,}\cdot\rangle+\langle\!\langle\cdot{,}\cdot\rangle\!\rangle.\)
- True
- False
MCQ 3
Every anti-symmetric bilinear form must be degenerate.
- True
- False
MCQ 4
The vector space of symmetric bilinear forms \(\mathbb{R}^2\times\mathbb{R}^2\to \mathbb{R}\) is \(3\)-dimensional.
- True
- False
MCQ 5
The vector space of alternating symmetric bilinear forms \(\mathbb{R}^2\times\mathbb{R}^2\to \mathbb{R}\) is \(1\)-dimensional.
- True
- False
MCQ 6
Let \(\vec a_1,\vec a_2\in\mathbb{K}^2\) denote the columns of a matrix \(\mathbf{A}\in M_{2,2}(\mathbb{K}).\) Then \(\langle\cdot{,}\cdot\rangle:\mathbb{K}^2\times\mathbb{K}^2\to\mathbb{K}\) defined by \(\langle \vec a,\vec b\rangle=\det(\mathbf{A})\) is a bilinear form.
- True
- False
MCQ 7
Consider a \(\mathbb{K}\)-vector space \(V.\) There does not exist a bilinear form on \(V\) which is both symmetric and alternating.
- True
- False
MCQ 8
Consider \(\mathbb{F}_2\) as an \(\mathbb{F}_2\) vector space. The bilinear form \(\mathbb{F}_2\times \mathbb{F}_2\to \mathbb{F}_2,\) \((x,y)\mapsto x y\mod 2\) is symmetric.
- True
- False
MCQ 9
Consider \(\mathbb{F}_2\) as an \(\mathbb{F}_2\) vector space. The bilinear form \(\mathbb{F}_2\times \mathbb{F}_2\to \mathbb{F}_2,\) \((x,y)\mapsto x y\mod 2\) is anti-symmetric.
- True
- False
MCQ 10
Consider \(\mathbb{F}_2\) as an \(\mathbb{F}_2\) vector space. The bilinear form \(\mathbb{F}_2\times \mathbb{F}_2\to \mathbb{F}_2,\) \((x,y)\mapsto x y\mod 2\) is alternating.
- True
- False