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3. (a) Let A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 2 & 1 \end{pmatrix}. i. Write down the characteristic polynomial of A. [6 marks] ii. Determine all the eigenvalues of A. [6 marks] iii. Determine the eigenspace $E_A(0)$. [5 marks] iv. What are the rank and nullity of A? Justify your answers briefly. [3 marks] (b) Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be defined by $T(x_1, x_2, x_3) = (0, x_1 + 2x_2, x_1 + x_3 - 2)$. Is T a linear mapping? Justify your answer. [4 marks] (c) Let $L: V \to W$ be a linear mapping between vector spaces V, W. Show that Ker(L) is a subspace of V. [5 marks] (d) Give an example of a non-zero matrix $A \in \mathbb{R}^{3 \times 3}$ for which 0 is an eigenvalue of algebraic multiplicity 3. [4 marks]

          3. (a) Let
A = \begin{pmatrix} 0 & 1 & 0 \\ 0 & 0 & 1 \\ 0 & 2 & 1 \end{pmatrix}.
i. Write down the characteristic polynomial of A. [6 marks]
ii. Determine all the eigenvalues of A. [6 marks]
iii. Determine the eigenspace $E_A(0)$. [5 marks]
iv. What are the rank and nullity of A? Justify your answers briefly. [3 marks]
(b) Let $T: \mathbb{R}^3 \to \mathbb{R}^3$ be defined by $T(x_1, x_2, x_3) = (0, x_1 + 2x_2, x_1 + x_3 - 2)$. Is T
a linear mapping? Justify your answer. [4 marks]
(c) Let $L: V \to W$ be a linear mapping between vector spaces V, W. Show that
Ker(L) is a subspace of V. [5 marks]
(d) Give an example of a non-zero matrix $A \in \mathbb{R}^{3 \times 3}$ for which 0 is an eigenvalue
of algebraic multiplicity 3. [4 marks]

3. (a) Let
A =
< p m a t r i x >
.
i. Write down the characteristic polynomial of A. [6 marks]
ii. Determine all the eigenvalues of A. [6 marks]
iii. Determine the eigenspace EA(0). [5 marks]
iv. What are the rank and nullity of A? Justify your answers briefly. [3 marks]
(b) Let T: ℝ^3 →ℝ^3 be defined by T(x1, x2, x3) = (0, x1 + 2x2, x1 + x3 - 2). Is T
a linear mapping? Justify your answer. [4 marks]
(c) Let L: V → W be a linear mapping between vector spaces V, W. Show that
Ker(L) is a subspace of V. [5 marks]
(d) Give an example of a non-zero matrix A ∈ℝ^3 × 3 for which 0 is an eigenvalue
of algebraic multiplicity 3. [4 marks]

Added by Mar P.

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