Question: True or False. (Please give a reason if True or a counterexample if False.)

a: If the kernel of a matrix \(A\) consists of the zero vector only, then the column vectors of \(A\) must be linearly independent.

b: If \(A\) is an invertible \(n \times n\) matrix, then the reduced row echelon form of \(A\) is \(I_{n}\)

c: If the \(4 \times 4\) matrix \(A\) has rank 4, then any linear system with coefficient matrix \(A\) will have a unique solution.

d: If a \(2 \times 2\) matrix \(P\) represents the orthogonal projection onto a line through the origin in \(\mathbb{R}^{2}\), then \(P\) is similar to the matrix

e: The map \(G: \mathcal{P}_{3} \rightarrow \mathcal{P}_{5}, f \mapsto 3 f(x) f^{\prime}(x)-2 f(x)\) is linear.

f: The map \(H: \mathcal{P}_{3} \rightarrow \mathbb{R}, f \mapsto \int_{-1}^{1} f(x) d x\) is linear.

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