MCQ 1
Let \(U=\operatorname{span}(\{f_1,f_2\})\) be a subspace of \(\mathsf C(\mathbb R,\mathbb R),\) where \[f_1(x) = \cos^2(x), f_2(x)=\sin^2(x).\] \(f(x) = \cos(2x)\) is an element of \(U.\)
- True
- False
MCQ 2
Let \(U=\operatorname{span}(\{f_1,f_2\})\) be a subspace of \(\mathsf C(\mathbb R,\mathbb R),\) where \[f_1(x) = \cos^2(x), f_2(x)=\sin^2(x).\] \(f(x) = 2\) is an element of \(U.\)
- True
- False
MCQ 3
The subset \[\left\{p(x)=\sum_{k=0}^na_kx^k\left|\sum_{k=1}^na_k=0\right\}\right.\] of \(\mathsf P_n(\mathbb{R})\) equipped with the usual addition and scalar multiplication is a subspace.
- True
- False
MCQ 4
The subset \[\left.\left\{p(x)=\sum_{k=0}^na_kx^k\right|a_k\in\mathbb Q, k=0,1,\ldots,n\right\}\] of \(\mathsf P_n(\mathbb{R})\) equipped with the usual addition and scalar multiplication is a subspace.
- True
- False
MCQ 5
If \(\operatorname{span}(\mathcal S)=\operatorname{span}(\mathcal T),\) then \(\mathcal S = \mathcal T.\)
- True
- False
MCQ 6
If \(\operatorname{span}(\mathcal S)\subset\operatorname{span}(\mathcal T),\) then \(\mathcal S\subset \mathcal T.\)
- True
- False
MCQ 7
If \(\mathcal S\subset \mathcal T,\) then \(\operatorname{span}(\mathcal S)\subset\operatorname{span}(\mathcal T).\)
- True
- False
MCQ 8
\(\operatorname{span}(\mathcal S)\cup\operatorname{span}(\mathcal T)=\operatorname{span}(\mathcal S\cup \mathcal T)\)
- True
- False
MCQ 9
\(\operatorname{span}(\mathcal S)\cap\operatorname{span}(\mathcal T)=\operatorname{span}(\mathcal S\cap \mathcal T)\)
- True
- False
MCQ 10
The map \(f:\mathsf P(\mathbb{R})\to\mathsf P(\mathbb{R})\) defined by \(p\mapsto p+\frac{\mathrm d}{\mathrm dx}p\) is invertible.
- True
- False
MCQ 11
The set \(W = \{\vec x \in \mathbb{K}^4 | x_1 - x_2 = x_3 + x_4 = 1\}\) is a subspace of \(\mathbb{K}^4.\)
- True
- False
MCQ 12
The set \(W = \{\vec x \in \mathbb{K}^4 | x_1 - x_2 = x_3 + x_4 = 0\}\) is a subspace of \(\mathbb{K}^4.\)
- True
- False
MCQ 13
If \(\vec a,\vec b\in\mathbb{K}^n,\) then the set \(W =\{\vec x \in \mathbb{K}^n | \vec a^T \vec x = \vec b^T \vec x = 0\}\) is a subspace of \(\mathbb{K}^n.\)
- True
- False
MCQ 14
If \(\vec a,\vec b\in\mathbb{K}^n,\) then the set \(W =\{\vec x \in \mathbb{K}^n | \vec a^T \vec x = \vec b^T \vec x = 1\}\) is a subspace of \(\mathbb{K}^n.\)
- True
- False
MCQ 15
If \(\vec a,\vec b\in\mathbb{K}^n,\) then the set \(W =\{\vec x \in \mathbb{K}^n | \vec a^T \vec x = \vec b^T\vec x = 1\}\cup \{0\}\) is a subspace of \(\mathbb{K}^n.\)
- True
- False