1. **Problem:** Examine if the function $f : \mathbb{R} \to \mathbb{R}$ defined by $f(x) = 2x + 1$ is one-one and onto. 2. **One-one (Injective) test:** A function is one-one if $f(x_1) = f(x_2)$ implies $x_1 = x_2$. Given $f(x_1) = 2x_1 + 1$ and $f(x_2) = 2x_2 + 1$, if $f(x_1) = f(x_2)$ then: $$2x_1 + 1 = 2x_2 + 1$$ Subtract 1 from both sides: $$2x_1 = 2x_2$$ Divide both sides by 2: $$x_1 = x_2$$ Hence, $f$ is one-one. 3. **Onto (Surjective) test:** A function is onto if for every $y \in \mathbb{R}$, there exists $x \in \mathbb{R}$ such that $f(x) = y$. Given $y$, solve for $x$: $$y = 2x + 1 \implies x = \frac{y - 1}{2}$$ Since $x$ is real for every real $y$, $f$ is onto. 4. **Conclusion:** $f$ is both one-one and onto. --- 5. **Problem:** Show that the set of integers $\mathbb{Z}$ with addition and multiplication forms a ring. 6. **Definition:** A ring is a set equipped with two operations (addition and multiplication) satisfying: - $(\mathbb{Z}, +)$ is an abelian group. - Multiplication is associative. - Multiplication distributes over addition. 7. **Verification:** - Addition in $\mathbb{Z}$ is associative, commutative, has identity 0, and every element has an inverse (negative integer). - Multiplication in $\mathbb{Z}$ is associative. - Distributive laws hold: $a(b+c) = ab + ac$ and $(a+b)c = ac + bc$ for all $a,b,c \in \mathbb{Z}$. Hence, $\mathbb{Z}$ with addition and multiplication is a ring. --- 8. **Problem:** Explain vector space over a field and give examples. 9. **Definition:** A vector space over a field $F$ is a set $V$ with two operations (vector addition and scalar multiplication) satisfying 8 axioms (closure, associativity, identity, inverses, distributivity, etc.). 10. **Example of vector space:** $\mathbb{R}^n$ over $\mathbb{R}$ with usual addition and scalar multiplication. 11. **Non-example:** The set of natural numbers $\mathbb{N}$ with usual addition and scalar multiplication is not a vector space because it lacks additive inverses. --- 12. **Problem:** Identify a reason why $\{(x,y,z) : x,y,z \in \mathbb{Q}\}$ is NOT a subspace over the real field $\mathbb{R}$. 13. **Reason:** For a subset to be a subspace over $\mathbb{R}$, it must be closed under scalar multiplication by any real number. Since $\mathbb{Q}$ is not closed under multiplication by irrational real numbers, scalar multiplication by $\mathbb{R}$ can produce vectors not in $\mathbb{Q}^3$. Hence, it is not a subspace over $\mathbb{R}$. --- 14. **Problem:** Define skew symmetric matrix and give an example of a $3 \times 3$ skew symmetric matrix. 15. **Definition:** A matrix $A$ is skew symmetric if $A^T = -A$. 16. **Example:** $$A = \begin{bmatrix} 0 & 2 & -1 \\ -2 & 0 & 3 \\ 1 & -3 & 0 \end{bmatrix}$$ --- 17. **Problem:** Evaluate eigenvalues and eigenvectors of matrix $$A = \begin{bmatrix} 1 & 6 & 1 \\ 0 & 2 & 4 \\ 0 & 0 & 3 \end{bmatrix}$$ 18. **Eigenvalues:** Since $A$ is upper triangular, eigenvalues are diagonal entries: $$\lambda_1 = 1, \quad \lambda_2 = 2, \quad \lambda_3 = 3$$ 19. **Eigenvectors:** For each eigenvalue $\lambda$, solve $(A - \lambda I)\mathbf{v} = 0$. - For $\lambda=1$: $$(A - I) = \begin{bmatrix} 0 & 6 & 1 \\ 0 & 1 & 4 \\ 0 & 0 & 2 \end{bmatrix}$$ From last row: $2v_3=0 \Rightarrow v_3=0$ From second row: $v_2 + 4v_3 = 0 \Rightarrow v_2=0$ From first row: $6v_2 + v_3=0$ (already zero) Eigenvector: $\mathbf{v}_1 = \begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}$ - For $\lambda=2$: $$(A - 2I) = \begin{bmatrix} -1 & 6 & 1 \\ 0 & 0 & 4 \\ 0 & 0 & 1 \end{bmatrix}$$ From last row: $v_3=0$ From second row: $0 \cdot v_2 + 4v_3=0 \Rightarrow$ no new info From first row: $-v_1 + 6v_2 + v_3=0 \Rightarrow -v_1 + 6v_2=0 \Rightarrow v_1=6v_2$ Eigenvector: $\mathbf{v}_2 = \begin{bmatrix} 6 \\ 1 \\ 0 \end{bmatrix}$ - For $\lambda=3$: $$(A - 3I) = \begin{bmatrix} -2 & 6 & 1 \\ 0 & -1 & 4 \\ 0 & 0 & 0 \end{bmatrix}$$ From last row: no info From second row: $-v_2 + 4v_3=0 \Rightarrow v_2=4v_3$ From first row: $-2v_1 + 6v_2 + v_3=0 \Rightarrow -2v_1 + 6(4v_3) + v_3=0 \Rightarrow -2v_1 + 25v_3=0 \Rightarrow v_1=\frac{25}{2}v_3$ Eigenvector: $\mathbf{v}_3 = \begin{bmatrix} \frac{25}{2} \\ 4 \\ 1 \end{bmatrix}$ --- **Final answers:** - $f(x) = 2x+1$ is one-one and onto. - $\mathbb{Z}$ with addition and multiplication forms a ring. - Vector space is a set with vector addition and scalar multiplication satisfying axioms; $\mathbb{R}^n$ is an example; $\mathbb{N}$ is not. - $\mathbb{Q}^3$ is not a subspace over $\mathbb{R}$ because it is not closed under scalar multiplication by irrationals. - Skew symmetric matrix satisfies $A^T = -A$; example given. - Eigenvalues of $A$ are 1, 2, 3; eigenvectors computed as above.