Problem 1

Question

determine whether the given set of vectors is linearly independent or linearly dependent in $\mathbb{R}^{n} .$ In the case of linear dependence, find a dependency relationship. $$\\{(3,6,9)\\}$$.

Step-by-Step Solution

Verified

Answer

The given set of vectors is linearly independent, as there is only a trivial linear combination (with scalar $k=0$) that results in the zero vector. There is no dependency relationship.

1Step 1: Check for trivial linear combination of the vector that results in the zero vector

: We have the vector $(3,6,9)$. Let's find a scalar, say $k$, that when multiplied by this vector results in the zero vector (0,0,0): \[ k(3,6,9) = (0,0,0) \]

2Step 2: Multiply the scalar to the vector and compare it to the zero vector

: Multiply the scalar by each component in the vector to obtain the resulting components: \[ (k \cdot 3, k \cdot 6, k \cdot 9) = (0,0,0) \] Now we compare each component to zero: \[ k \cdot 3 = 0\\ k \cdot 6 = 0\\ k \cdot 9 = 0 \]

3Step 3: Determine the scalar and check if it's a non-trivial linear combination

: We know that if $k = 0$, these equations will hold true. Therefore, we have a trivial linear combination: \[ 0 \cdot (3,6,9) = (0,0,0) \] However, let's check if there is any non-trivial linear combination. From the first equation, we get: \[ k = \frac{0}{3} = 0 \] Since $k = 0$, we have only a trivial linear combination of the given set of vectors that results in the zero vector. Therefore, the set of vectors is linearly independent. In this case, there is no dependency relationship.

Key Concepts

Vectors in $\mathbb{R}^n$Linear DependenceZero Vector

Vectors in $\mathbb{R}^n$

Vectors in $\mathbb{R}^n$ are fundamental in understanding the structure of our mathematical space. $\mathbb{R}^n$ denotes the n-dimensional real coordinate space. It includes all possible n-tuples of real numbers. For example, a vector in $\mathbb{R}^3$ is represented as $(x_1, x_2, x_3)$ where $x_1, x_2,$ and $x_3$ are real numbers.

These vectors are used to describe points, directions, or various physical quantities in three-dimensional space, among others. Any vector in $\mathbb{R}^n$ can be written as $(a_1, a_2, ..., a_n)$, where each $a_i$ is a component of the vector.

Magnitude: The length of the vector which can be calculated using the formula $\sqrt{a_1^2 + a_2^2 + \ldots + a_n^2}$.
Direction: Gives the vector its position or path in space relative to a fixed point.

Understanding vectors in $\mathbb{R}^n$ is essential for engaging with more advanced topics like linear transformations and matrices.

Linear Dependence

Linear dependence is an intrinsic property of sets of vectors. A set of vectors is considered linearly dependent if one or more vectors in the set can be expressed as a linear combination of the others. This means that to create one vector in the set, you could add together, or scale, the others.

Here’s the intuitive way to think about it: if you have a group of friends and some are always tagging along without contributing their own unique paths (or directions), they could be considered dependent—much like some vectors that rely on combinations of others.

In problem-solving, you know a set of vectors $\{v_1, v_2, \ldots, v_k\}$ is linearly dependent if there exist scalars $c_1, c_2, \ldots, c_k$, not all zero, such that:\[c_1v_1 + c_2v_2 + \cdots + c_kv_k = 0\] A quick tip: If a set includes more vectors than the number of dimensions they exist in, then they are definitely dependent. This is known as the "rank" problem where vectors "overcrowd" the space and start to overlap.

Zero Vector

The zero vector is a special type of vector that has zero magnitude and no particular direction. In $\mathbb{R}^n$, it is denoted as $(0, 0, 0, \ldots, 0)$ with n components, each being zero. The zero vector is unique because:

Adding it to any other vector doesn’t change the vector, i.e., $v + 0 = v$.
It acts as an additive identity in vector spaces.

In terms of linear independence and dependence, the zero vector is especially pivotal. Any time a vector set includes the zero vector, it is automatically linearly dependent. This is due to the fact that the zero vector can be created by any scalar multiple, such as $k \cdot 0$, fulfilling that equation of dependence.

The zero vector is crucial for understanding vector spaces and related algebraic structures, as it provides a baseline or reference point in the vector world.

Problem 1