(Solved): (a) Consider two vector spaces \( V \) and \( W \). Let \( T: V \rightarrow W ...

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(a) Consider two vector spaces \( V \) and \( W \). Let \( T: V \rightarrow W \) be a linear map. Define the kernel of \( T \) and show that it is a subspace of \( V \). (b) Now let \( S: \mathbb{R}^{4} \rightarrow \mathbb{R}^{3} \), with \( S(x, y, w, z)=(x+w,-x+y-z, y-z+w) \) and \( T: \mathbb{R}^{3} \rightarrow \mathbb{R}^{2} \), with \( T(x, y, z)=(x-2 y+z,-2 x+4 y-2 z) \). i. Find a matrix representation of the maps \( S \) and \( T \) with respect to the standard bases on \( \mathbb{R}^{n} \). Using these, show that the matrix of \( T \circ S \) is given by \[ A_{T \circ S}=\left(\begin{array}{cccc} 3 & -1 & 1 & 2 \\ -6 & 2 & -2 & -4 \end{array}\right) . \] ii. Find \( \operatorname{ker}(S) \) and \( \operatorname{ker}(T \circ S) \). Find the nullity of \( S \) and \( T \circ S \), denoted \( n(S) \) and \( n(T \circ S) \). iii. State the rank-nullity theorem and use it to find the rank of \( S \) and \( T \circ S \), denoted \( r(S) \) and \( r(T \circ S) \). (c) Consider arbitrary linear maps \( S: V \rightarrow W, T: W \rightarrow U \) and \( T \circ S: V \rightarrow U \), where \( U, V \) and \( W \) are finite-dimensional vector spaces. Prove that \( \operatorname{ker}(S) \subseteq \operatorname{ker}(T \circ S) \). Hence prove that \( n(S) \leq n(T \circ S) \).