Injective and surjective maps
Task number: 4460
Denote by \(P\) the space of real polynomials of degree at most 2.
Decide which of the following linear mappings are injective and which are surjective:
Variant
\(f:\mathbb{R}^{2\times2} \to \mathbb{R}^3\) given by \(f\begin{pmatrix} a & b \\ c & d \end{pmatrix} = (a + b + c, a + b, a)^T\)Variant
\(f:\mathbb{R}^{2\times2} \to \mathbb{R}^4\) given by \(f\begin{pmatrix} a & b \\ c & d \end{pmatrix} = (a + b + c + d, a + b + c, a + b, a)^T\)Variant
\(f:\mathbb{R}^{2\times2} \to P\) given by \(f\begin{pmatrix} a & b \\ c & d \end{pmatrix} = (a + b)x^2 + (c + d)x + c\)Variant
\(f:P \to \mathbb{R}^4\) given by \(f(ax^2 + bx + c) = (a - b + c, b + c, a + 2c, a - c)^T\)Variant
\(f:P \to \mathbb{R}^3\) given by \(f(ax^2 + bx + c) = (a + b, 2b - c, a - b + c)^T\)Variant
\(f:P \to \mathbb{R}^3\) given by \(f(ax^2 + bx + c) = (a + b, 2b - c, a - b + 2c)^T\)
