If \(\mathbb K\) is a field, then it is a vector space over itself.

• True
• False

The set of polynomial functions of degree \(2\) in one variable with coefficients in \(\mathbb R\) equipped with \(+_{\mathsf P(\mathbb R)}\) and \(\cdot_{\mathsf P(\mathbb R)}\) forms a vector space.

• True
• False

The set \(\{f:\mathbb R\to \mathbb R\, |\, f(3)=2\}\) equipped with the usual addition and scalar multiplication for functions forms a vector space.

• True
• False

The set \(\{f:\mathbb R\to \mathbb R\, |\, f(2)=0\}\) equipped with the usual addition and scalar multiplication for functions forms a vector space.

• True
• False

If \(\mathbf A\in M_{n,n}(\mathbb K)\) is such that \[\mathbf{1}_n + \mathbf A + \ldots + \mathbf A^m = \mathbf{0}_n\] for some \(m\in\mathbb N,\) then \(\mathbf A\) admits an inverse.

• True
• False

The map \(f:\mathsf P(\mathbb{R})\to\mathsf P(\mathbb{R})\) defined by \(p\mapsto p+\frac{\mathrm d^2}{\mathrm dx^2}p\) is linear.

• True
• False

If \(\mathbf A,\mathbf B,\mathbf C\in M_{n,n}(\mathbb K)\) are is such that \(\mathbf A\mathbf B = \mathbf A\mathbf C = \mathbf{1}_{n},\) then \(\mathbf B = \mathbf C.\)

• True
• False

If \(\mathbf A \in M_{n,n}(\mathbb K)\) is such that \(\mathbf A^3 = \mathbf{1}_{n}\) and \(\mathbf A,\mathbf A^2,\ne\mathbf{1}_{n},\) then \(\mathbf A^{-1} = \mathbf A^{3k+1}\) for any natural number \(k.\)

• True
• False

If \(\mathbf A,\mathbf B \in M_{n,n}(\mathbb K)\) are both invertible, then \((\mathbf A\mathbf B)^{-1} = \mathbf A^{-1}\mathbf B^{-1}.\)

• True
• False

If \(\mathbf A \in M_{n,n}(\mathbb K)\) is invertible, then \(\mathbf{A}^T\) is also invertible and \((\mathbf A^T)^{-1}=(\mathbf A^{-1})^T.\)

• True
• False

Rotation about the origin, regarded as a map from the vector space \(\mathbb{R}^2\) to itself, is a linear map.

• True
• False

Reflection about a line through the origin in \(\mathbb{R}^2,\) regarded as a map from the vector space \(\mathbb{R}^2\) to itself, is a linear map.

• True
• False

Reflection about a line in \(\mathbb{R}^2,\) regarded as a map from the vector space \(\mathbb{R}^2\) to itself, is a linear map.

• True
• False

Multiplication by a scalar \(c\in\mathbb{R},\) regarded as a map from the vector space \(\mathbb{R}^2\) to itself, is a linear map.

• True
• False

Projection onto the \(x\)-axis, regarded as a map from the vector space \(\mathbb{R}^2\) to itself, is a linear map.

• True
• False