Exercises: Dependence and independence. Bases

Two surveillance drones move through space along straight trajectories defined by the vectors $\vec{u} = (2, -4, 6)$ and $\vec{v} = (-1, 2, -3)$. Determine whether the drones move in parallel directions by analysing the linear dependence of their direction vectors.

In a video game graphics engine, the position of an object is defined by a linear combination of two basic vectors: $\vec{a} = (1, 0, 2)$ and $\vec{b} = (0, -1, 3)$. Calculate the coordinates of the resulting vector $\vec{w}$ if the following operation is applied:

Given the vectors $\vec{u} = (1, 2, 3)$, $\vec{v} = (2, 0, 1)$ and $\vec{w} = (3, 2, 4)$, prove without performing complex calculations (by observing the relationship between their components) that the set $\{\vec{u}, \vec{v}, \vec{w}\}$ is linearly dependent.

A laser sensor system requires two beams of light to be parallel. The first beam follows the direction of vector $\vec{d_1} = (3, k, -1)$ and the second follows vector $\vec{d_2} = (6, -4, -2)$. Find the value of parameter $k$ so that the vectors are linearly dependent and, therefore, the beams are parallel.

Check whether the vectors $\vec{e_1} = (1, 0, 0)$, $\vec{e_2} = (1, 1, 0)$ and $\vec{e_3} = (1, 1, 1)$ can form a basis of $\mathbb{R}^3$. To do this, justify whether they are linearly independent using the determinant of the matrix they form.

Consider the points in space $A(1, 2, -1)$, $B(2, 3, 1)$ and $C(3, 4, 3)$.

1. a) Find the vectors $\vec{AB}$ and $\vec{AC}$.
2. b) From their components, decide whether points $A, B$ and $C$ are collinear.

If a vector $\vec{v}$ has coordinates $(3, -2, 5)$ with respect to the standard basis $\{\vec{i}, \vec{j}, \vec{k}\}$, express it as a linear combination of these unit vectors.

State with reasons whether the following statement is true or false: "In three-dimensional space $\mathbb{R}^3$, any set of four non-zero vectors is always linearly dependent." Give a simple example to support your answer.

To check whether the vectors $\vec{u} = (1, 2, 0)$, $\vec{v} = (2, 1, 0)$, and $\vec{w} = (3, 3, 0)$ form a basis of $\mathbb{R}^3$, Marcos writes the following reasoning on his exam:

"I check whether they are pairwise proportional:

• $\vec{u}$ and $\vec{v}$ are not proportional because $\frac{1}{2} \neq \frac{2}{1}$.

• $\vec{u}$ and $\vec{w}$ are not proportional because $\frac{1}{3} \neq \frac{2}{3}$.

• $\vec{v}$ and $\vec{w}$ are not proportional because $\frac{2}{3} \neq \frac{1}{3}$.

Since no pair of vectors has the same direction, the three vectors are linearly independent and, therefore, constitute a basis of $\mathbb{R}^3$".

1. a) Identify and explain in detail the conceptual error Marcos makes.
2. b) Explain, without performing algebraic calculations and looking only at the third component of the vectors, why it was obvious from the beginning that they could not form a basis of three-dimensional space.
3. c) Apply the calculation of the determinant $\det(\vec{u}, \vec{v}, \vec{w})$ using Sarrus' rule to provide the formal analytical justification that refutes Marcos's conclusion.

Design three vectors in three-dimensional space, $\vec{u}$, $\vec{v}$, and $\vec{w}$, that simultaneously satisfy the following three conditions:

1. None of the three vectors has zero components (no component can be $0$).

2. No pair of vectors is proportional to each other (that is, no two vectors have the same direction).

3. The set $\{\vec{u}, \vec{v}, \vec{w}\}$ is linearly dependent.

To achieve this:

1. a) Explain the strategy you followed to construct them from the definition of a linear combination.
2. b) Write the numerical components of your three vectors.
3. c) Formally verify that they are dependent by calculating the determinant formed by their components and checking that its value is exactly $0$.

Given the vectors $\vec{u} = (1, 2, -1)$ and $\vec{v} = (0, 1, 2)$, determine the value of the real parameter $k$ such that vector $\vec{w} = (2, k, 0)$ is a linear combination of $\vec{u}$ and $\vec{v}$. What geometrical relationship do the three vectors have for that value of $k$?

Study the linear dependence or independence of the set of vectors $S = \{\vec{a}, \vec{b}, \vec{c}\}$ according to the values of the parameter $m \in \mathbb{R}$, where $\vec{a} = (m, 1, 1)$, $\vec{b} = (1, m, 1)$, and $\vec{c} = (1, 1, m)$.

Give a reasoned proof that if vectors $\{\vec{u}, \vec{v}, \vec{w}\}$ are linearly independent, then vectors $\{\vec{u} + \vec{v}, \vec{v} + \vec{w}, \vec{w} + \vec{u}\}$ are also linearly independent and, therefore, form a basis of $\mathbb{R}^3$.

A surveillance drone moves along straight trajectories defined by vectors. If the direction vectors of three drones are $\vec{d_1} = (1, 3, a)$, $\vec{d_2} = (4, 1, 2)$ and $\vec{d_3} = (a, -1, 0)$, calculate the values of $a$ for which the trajectories of the three drones would be contained in planes parallel to each other (that is, the vectors are coplanar).

Consider the points in space $A(1, 0, 0)$, $B(0, 2, 1)$, and $C(2, 0, m)$.

1. a) Find the vector $\vec{AB}$ and the vector $\vec{AC}$.
2. b) Determine the value of $m$ such that the vector $\vec{v} = (3, -3, -3)$ is linearly dependent on $\vec{AB}$ and $\vec{AC}$.
3. c) For the value of $m$ found, write $\vec{v}$ as a linear combination of $\vec{AB}$ and $\vec{AC}$.

Given the vectors of the standard basis $\{\vec{i}, \vec{j}, \vec{k}\}$, the following vectors are defined:

Justify whether the set $\{\vec{u}, \vec{v}, \vec{w}\}$ is a basis of $\mathbb{R}^3$. If a vector $\vec{x}$ has coordinates $(1, 2, 3)$ in the standard basis, what will its coordinates be in the basis $B' = \{\vec{u}, \vec{v}, \vec{w}\}$?

Determine whether the points $P(1, 2, -1)$, $Q(3, 1, 1)$, $R(0, 5, -2)$ and $S(k, 0, 4)$ are coplanar depending on the value of $k$. To carry out this analysis, use the linear dependence of the vectors originating at $P$ with endpoints at the other points.

Consider the vectors $\vec{u} = (1, -1, 2)$, $\vec{v} = (2, 0, 3)$, and $\vec{w} = (0, 2, -1)$. We want to find out whether $\vec{w}$ is coplanar with $\vec{u}$ and $\vec{v}$. Two classmates propose solving it using different approaches:

• Mateo's method: Directly calculate the determinant $\det(\vec{u}, \vec{v}, \vec{w})$ using Sarrus' rule.

• Elena's method: Set up the vector equation $\vec{w} = a\cdot\vec{u} + b\cdot\vec{v}$ as a system of three equations with two unknowns ($a$ and $b$), solve it using two of them, and check whether the third is satisfied.

1. a) Apply Mateo's method to determine whether $\vec{w}$ is coplanar with $\vec{u}$ and $\vec{v}$.
2. b) Apply Elena's method to find, if they exist, the scalars $a$ and $b$ that express $\vec{w}$ as a linear combination of $\vec{u}$ and $\vec{v}$.
3. c) Compare both procedures: if in a test you are only asked "are these three vectors coplanar?", which one would you choose and why? How would your choice change if you were also asked "express vector $\vec{w}$ in terms of $\vec{u}$ and $\vec{v}$"?

A reconnaissance drone unit of the BRIF (Forest Fire Reinforcement Brigades) in the Sierra de Gredos uses three elementary displacement vectors calibrated in its engines (measured in kilometres along the East, North, and Altitude axes):

$\vec{d}_1 = (2, 1, 0,5)$, $\vec{d}_2 = (1, 3, 1)$, and $\vec{d}_3 = (4, -1, k)$, where $k$ is a vertical response parameter adjustable via software.

1. a) For the drone to be able to manoeuvre to any point in three-dimensional space, its three elementary movements must form a basis of $\mathbb{R}^3$. Determine the critical value of $k$ for which the movements would be confined to a single plane (that is, they would be linearly dependent), preventing full control of the drone.
2. b) If the technicians set $k = 1\text{ km}$, justify with reasoning why the three vectors form a basis $B = \{\vec{d}_1, \vec{d}_2, \vec{d}_3\}$.
3. c) To reach a fire hotspot located at relative position $\vec{p} = (7, 3, 2,5)\text{ km}$, calculate how many units of each elementary movement $(a, b, c)$ the pilot must program; that is, calculate the coordinates of $\vec{p}$ in basis $B$.

A team of engineers is designing the navigation system for a satellite. In order for the satellite to orient itself in any direction in three-dimensional space, it needs three thrusters whose thrust directions are linearly independent. Three thrusters defined by the vectors $\vec{u} = (1, m, 1)$, $\vec{v} = (m, 1, 1)$ and $\vec{w} = (1, 1, m)$ have been installed.

Determine, giving reasons, for which values of the real parameter $m$ the propulsion system would not be able to manoeuvre in all of space (that is, when the vectors are linearly dependent). In the case where $m=1$, explain geometrically what happens to the thrust directions and whether they form a basis for $\mathbb{R}^3$.

Given the vectors in space $\vec{a} = (1, 2, -1)$, $\vec{b} = (0, 1, 2)$, and $\vec{c} = (k, 1, -3)$, answer the following:

1. a) Find the value of $k$ so that the vector $\vec{v} = (3, 4, -7)$ can be expressed as a linear combination of $\vec{a}$, $\vec{b}$, and $\vec{c}$ such that the three vectors are linearly dependent.

1. b) If $k = 2$, do they form a basis of $\mathbb{R}^3$? If so, calculate the coordinates of the vector $\vec{w} = (1, 1, 1)$ with respect to this basis $B = \{ \vec{a}, \vec{b}, \vec{c} \}$.

In open-world video game development, the position of a character is defined using a non-standard basis $B' = \{ \vec{u_1}, \vec{u_2}, \vec{u_3} \}$. We know that these vectors are related to the standard basis $B = \{ \vec{i}, \vec{j}, \vec{k} \}$ as follows:

$\vec{u_1} = \vec{i} + \vec{j}$

$\vec{u_2} = \vec{j} + \vec{k}$

$\vec{u_3} = \vec{i} + \vec{k}$

1. a) Show that $B'$ is a basis for $\mathbb{R}^3$.

1. b) If an object in the game has coordinates $(2, -4, 6)$ in the standard basis, determine its coordinates in the basis $B'$.

Study the linear dependence or independence of the following vectors according to the values of the parameters $a$ and $b$:

$\vec{u} = (a, 1, 0)$

$\vec{v} = (1, a, 1)$

$\vec{w} = (0, 1, b)$

Determine for which relationship between $a$ and $b$ the vectors are coplanar. Is there any value of $a$ for which the vectors are linearly dependent regardless of the value of $b$? Justify your answer.

Let $\{\vec{u}, \vec{v}, \vec{w}\}$ be a basis of $\mathbb{R}^3$. Consider the vectors:

$\vec{x} = \vec{u} + \vec{v}$

$\vec{y} = \vec{v} + \vec{w}$

$\vec{z} = \vec{w} + \lambda\vec{u}$

1. a) For which values of $\lambda$ does the set $\{\vec{x}, \vec{y}, \vec{z}\}$ not form a basis of $\mathbb{R}^3$?

1. b) For $\lambda = 0$, express the vector $\vec{w}$ as a linear combination of $\vec{x}$, $\vec{y}$ and $\vec{u}$.

An architect designs a tensile structure using three cables that converge at an anchor point $P$. The directions of the cables are given by the vectors $\vec{v_1} = (1, 1, a)$, $\vec{v_2} = (a, 0, 1)$ and $\vec{v_3} = (2, 1, 1)$.

1. a) If the anchor is to be stable, the cables cannot all lie in the same plane. Find the values of $a$ that would cause the structure to collapse because the vectors are linearly dependent.

1. b) For $a = 2$, a total force of $\vec{F} = (5, 2, 4)$ is applied at point $P$. Determine whether this force can be uniquely decomposed as a sum of tensions in the directions of the cables and, if so, find this decomposition.

Consider the points $A(1, 0, 1)$, $B(0, 2, 1)$ and $C(2, 1, 0)$.

1. a) Find a point $D$ such that vector $\vec{AD}$ is linearly dependent on vector $\vec{BC}$ and has a magnitude of 6. Is the solution unique?

1. b) Determine whether point $E(5, -1, 4)$ belongs to the plane defined by vectors $\vec{AB}$ and $\vec{AC}$. Justify your answer using the concepts of linear dependence and basis.

Prove that if three vectors $\vec{u}, \vec{v}, \vec{w}$ are linearly independent, then vectors $\vec{u} - \vec{v}$, $\vec{v} - \vec{w}$ and $\vec{w} - \vec{u}$ are always linearly dependent.

Give a geometrical interpretation of this result, considering the vectors as the sides of a closed figure in space.

Consider the basis $B = \{\vec{u}, \vec{v}, \vec{w}\}$ of $\mathbb{R}^3$ formed by the vectors $\vec{u} = (10, 0, 0)$, $\vec{v} = (1, 10, 0)$, and $\vec{w} = (1, 1, 10)$, and the target vector $\vec{x} = (12, 21, 30)$. We seek the coordinates $(a, b, c)_B$ such that $\vec{x} = a\cdot\vec{u} + b\cdot\vec{v} + c\cdot\vec{w}$.

1. a) Estimate: Before solving any numerical system, analyse the structure of the basis vectors (observe the position of the zeros and the dominant values $10$):

• Which vector of the basis is the only one capable of contributing to the third component ($z$)? Estimate the exact value that $c$ must have.

• Knowing that this value of $c$ already contributes a fixed amount to the second component ($y$), around what value should coordinate $b$ lie? Will it be greater or less than $2,1$?

• Reasoning analogously with the first component ($x$), estimate whether coordinate $a$ will be greater or less than $1$.

1. b) Calculate: Set up the linear system corresponding to $\vec{x} = a\cdot\vec{u} + b\cdot\vec{v} + c\cdot\vec{w}$ and calculate the exact values of $a$, $b$, and $c$.
2. c) Assess: Compare your qualitative estimates with the exact results and explain why the triangular arrangement of this basis causes a "cascade effect" that successively corrects each coordinate from bottom to top.

Consider the set of vectors in $\mathbb{R}^3$ formed by $\vec{u} = (1, 1, 0)$, $\vec{v} = (1, 0, 1)$, and $\vec{w} = (0, 1, 1)$.

1. a) Prove that $B = \{\vec{u}, \vec{v}, \vec{w}\}$ is a basis of $\mathbb{R}^3$ by calculating the volume of the parallelepiped they determine from their scalar triple product or determinant.
2. b) A student firmly states: "Since $B$ is not the standard basis, it is impossible for a non-zero vector to have exactly the same components in the standard basis as coordinates in basis $B$". To check whether they are right, set up the vector equation that translates this condition:

Transform this equality into a homogeneous linear system, solve it with reasoning, and conclude whether the student's statement is true or false.

1. c) Geometrically interpret the set of vectors that satisfy the above condition, indicating what shape they form in space and what relationship they have with the vectors of basis $B$.