Algebra and Geometry | Part IA, 2003

Let α\alpha be a linear map

α:R3R3.\alpha: \mathbb{R}^{3} \rightarrow \mathbb{R}^{3} .

Define the kernel KK and image II of α\alpha.

Let yR3\mathbf{y} \in \mathbb{R}^{3}. Show that the equation αx=y\alpha \mathbf{x}=\mathbf{y} has a solution xR3\mathbf{x} \in \mathbb{R}^{3} if and only if yI\mathbf{y} \in I

Let α\alpha have the matrix

(11t0t2b1t0)\left(\begin{array}{ccc} 1 & 1 & t \\ 0 & t & -2 b \\ 1 & t & 0 \end{array}\right)

with respect to the standard basis, where bRb \in \mathbb{R} and tt is a real variable. Find KK and II for α\alpha. Hence, or by evaluating the determinant, show that if 0<b<20<b<2 and yI\mathbf{y} \in I then the equation αx=y\alpha \mathbf{x}=\mathbf{y} has a unique solution xR3\mathbf{x} \in \mathbb{R}^{3} for all values of tt.

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