LQ3 Mock

Tags: #◇CSMATH2 #◇Linear Algebra #◇Computational Mathematics #◇Matrices #◇Matrix Operations #◇Computer Science #◇Vector Spaces #◇Linear Transformations

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◇Overview

Instructions: Please write down your answers and solve the problems on paper. e

◇Details / Theory

Instructions: Please write down your answers and solve the problems on paper. e

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Part I: Multiple Choice (20 Items)

Select the best answer for each question.

1. Let $B_1$ and $B_2$ denote two bases of an $n$-dimensional space $V$. Which of the following statements is always true?
   A) There is no common vector in $B_1$ and $B_2$.
   B) If there exists a common vector of $B_1$ and $B_2$, then $B_1 = B_2$.
   C) $B_1$ and $B_2$ contain the same number of vectors.
   D) Some vectors in $B_1$ are also in $B_2$.

2. If $B = \{u_1, u_2, ..., u_n\}$ is a basis for a vector space $V$, then every vector in $V$ can be written uniquely as a linear combination of the vectors in $B$.
   A) TRUE
   B) FALSE

3. Let $S = \{u_1, u_2, ..., u_n\}$ be a linearly independent set in an $n$-dimensional vector space $V$. Which of the following is always true?
   A) There exists a spanning set $S'$ such that $S' \subset S$.
   B) $S$ can be extended to a larger linearly independent set.
   C) $S$ is a basis for $V$.
   D) $S$ does not span $V$.

4. If the set $S$ is a linearly dependent subset of a vector space $V$, which of the following is always true?
   A) One of the vectors in $S$ can be expressed as a linear combination of the other vectors in $S$.
   B) $S$ contains a basis for $V$.
   C) One of the vectors in $S$ is the zero vector.
   D) $S$ spans $V$.

5. Let $V$ be a vector space of dimension $n$, and let $S \subset V$ be a set of $m$ vectors with $m < n$. What can we conclude?
   A) $S$ is a basis for $V$.
   B) $S$ must be linearly dependent.
   C) $S$ cannot span $V$.
   D) $S$ may or may not span $V$.

6. Which of the following is a linearly independent subset of $\mathbb{R}^2$?
   A) $\{(1,1), (2,2)\}$
   B) $\{(1,1)\}$
   C) $\{(0,0)\}$
   D) $\{(1,2), (0,0)\}$

7. Consider the range of a linear transformation given by Range $L = \{ \begin{bmatrix} a \\ a-b \\ b \end{bmatrix} \mid a, b \in \mathbb{R} \}$. Which of the following spans Range $L$?
   A) $\{ \begin{bmatrix} a \\ a-b \\ b \end{bmatrix} \}$
   B) $S = \{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix} \}$
   C) $S = \{ \begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}, \begin{bmatrix} 0 \\ -1 \\ 1 \end{bmatrix} \}$
   D) $S = \{ \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \begin{bmatrix} 0 \\ -1 \\ 0 \end{bmatrix} \}$

8. Using the transformation in Question #7, which of the following is true?
   A) $L$ is onto (assuming codomain $\mathbb{R}^3$).
   B) $dim(\text{Range } L) = 2$.
   C) $L$ is one-to-one (assuming domain $\mathbb{R}^3$).
   D) Both B and C.

9. Consider $L: \mathbb{R}^2 \to \mathbb{R}^3$ defined by $L(x,y) = (x+y, x-y, 0)$. Which of the following is the Range of $L$?
   A) $\{(0,0,0)\}$
   B) Span${(1,1,0)}$
   C) Span${(1,1,0), (1,-1,1)}$
   D) Span${(1,1,0), (1,-1,0)}$

10. Using the transformation in Question #9, which of the following is the Kernel of $L$?
    A) $\{(0,0)\}$
    B) $\{(1,1), (1,-1)\}$
    C) $\{(2x, 0) \mid x \in \mathbb{R}\}$
    D) $\{(1,0), (0,1)\}$

11. Which of the following is true about a linear transformation $L: V \to W$?
    A) The range of $L$ contains $W$.
    B) If Ker $L = \{0_V\}$, then $L$ is one-to-one.
    C) When $L$ is one-to-one then it is onto.
    D) Both A and B.

12. Which of the following is a linear transformation that is both one-to-one and onto?
    A) $L: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $L(x,y) = (0,0)$
    B) $L: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $L(x,y) = (0,y)$
    C) $L: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $L(x,y) = (-x, -y)$
    D) $L: \mathbb{R}^2 \to \mathbb{R}^2$ defined by $L(x,y) = (x, 2x)$

13. A function $L: V \to W$ is a linear transformation if and only if:
    A) $L(ru + sv) = rL(u) + sL(v)$ for all $u, v \in V$ and scalars $r, s$.
    B) $L$ is one-to-one and onto.
    C) The image of $L$ is a line.
    D) $L(0_V) \neq 0_W$.

14. Let $L: \mathbb{R}^2 \to \mathbb{R}^2$ be a linear transformation where $L([1, 0]^T) = [2, 1]^T$ and $L([0, 1]^T) = [0, -1]^T$. What is $L([3, 2]^T)$?
    A) $[6, 1]^T$
    B) $[2, 0]^T$
    C) $[6, 2]^T$
    D) $[0, 0]^T$

15. Which of the following is not a linear transformation from $\mathbb{R}^3$ to $\mathbb{R}^2$?
    A) $L(a,b,c) = (2a, c-b)$
    B) $T(a,b,c) = (a+b, a-c)$
    C) $T(a,b,c) = (a, b)$
    D) $L(a,b,c) = (a^2, 0)$

16. What is the dimension of the vector space $\mathbb{R}^4$?
    A) 2
    B) 3
    C) 4
    D) Infinite

17. The set of vectors $\{(1,0,0), (0,1,0), (0,0,1), (1,1,1)\}$ in $\mathbb{R}^3$ is:
    A) Linearly Independent
    B) A Basis for $\mathbb{R}^3$
    C) Linearly Dependent
    D) None of the above

18. If $L: V \to W$ is a linear transformation and $dim(V) = 5$ and $dim(\text{Range } L) = 3$, what is $dim(\text{Ker } L)$?
    A) 2
    B) 3
    C) 5
    D) 8

19. Which of the following sets spans $\mathbb{R}^2$?
    A) $\{(1,0)\}$
    B) $\{(1,1), (2,2)\}$
    C) $\{(1,0), (0,1), (2,3)\}$
    D) $\{(0,0), (1,1)\}$

20. Let $L(x) = Mx$ where $M$ is a $3 \times 2$ matrix. The domain of $L$ is:
    A) $\mathbb{R}^2$
    B) $\mathbb{R}^3$
    C) A matrix space
    D) The empty set

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Part II: Problem Solving (5 Items)

Show your complete solution for each item.

21. Linear Independence & Basis

Determine if the set $S = \left\{ (1, 2), (3, 4) \right\}$ is a basis for $\mathbb{R}^2$.

(Hint: You must check two conditions: Linear Independence and Spanning).

22. Extending a Basis

Given the set $A = \{ (1, 1, 0), (1, 0, 1) \}$ in $\mathbb{R}^3$.

Find a third vector $v_3$ such that $\{ (1, 1, 0), (1, 0, 1), v_3 \}$ forms a basis for $\mathbb{R}^3$. Show that your chosen vector makes the set linearly independent.

23. Linearity Test

Let $L: \mathbb{R}^2 \to \mathbb{R}$ be defined by $L(x, y) = 3x + y + 1$.

Is $L$ a linear transformation? Prove your answer using the definition of a linear transformation or a property of the zero vector.

24. Kernel and Range

Let $L: \mathbb{R}^3 \to \mathbb{R}^2$ be defined by:

$L\left(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\right) = \begin{bmatrix} x + y \\ z \end{bmatrix}$
a) Find the Kernel of $L$ (describe the set of vectors).
b) Find the dimension of the Range of $L$.

25. Transformation of a Vector

Let $L: \mathbb{R}^2 \to \mathbb{R}^2$ be a linear transformation such that:

$L(1, 0) = (2, 3)$

$L(0, 1) = (-1, 4)$

Find $L(5, -2)$.

◇Common Pitfalls

Confusing linear independence with spanning — a set can span a space without being linearly independent (redundant vectors) Forgetting to row-reduce before determining basis, dimension, kernel, or range — always reduce to RREF first Misapplying the rank-nullity theorem — $\text{rank}(T) + \text{nullity}(T) = \dim(\text{domain})$, not $\dim(\text{codomain})$ Confusing kernel (maps to zero) with range (all possible outputs) — kernel tests one-to-one, range tests onto Arithmetic errors in linear transformation problems — always verify by substituting back into the original transformation

◇Summary

TL;DR — This mock exam covers linear independence, basis and dimension, linear transformations, kernel and range, and the rank-nullity theorem. 20 multiple-choice + 5 problem-solving items. Key strategy: row-reduce first, then read off pivot/free columns.

◇Links

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◇References & Navigation

• Related Notes: ◇Linear Equations and Matrices