Question

We often write vectors in \(\mathbb{R}^{n}\) as rows. In each case determine if the given vectors span \(\mathbb{R}^{4}\). Support your answer. $$ \begin{array}{l} \text { a. }\\{(1,1,1,1),(0,1,1,1),(0,0,1,1),(0,0,0,1)\\} \\ \text { b. }\\{(1,3,-5,0),(-2,1,0,0),(0,2,1,-1), \\ \quad(1,-4,5,0)\\} \end{array} $$

Short answer

Set (a) spans \( \mathbb{R}^4 \); set (b) doesn't.

Step-by-step solution

Define Spanning Vectors

Vectors span \( \mathbb{R}^4 \) if they can generate any vector in \( \mathbb{R}^4 \) as a linear combination. For four vectors to span \( \mathbb{R}^4 \), they must be linearly independent, meaning the determinant of their coefficient matrix is non-zero.

Evaluate Set a

Given vectors: \((1,1,1,1), (0,1,1,1), (0,0,1,1), (0,0,0,1)\). Construct the matrix: \[ A = \begin{pmatrix} 1 & 1 & 1 & 1 \ 0 & 1 & 1 & 1 \ 0 & 0 & 1 & 1 \ 0 & 0 & 0 & 1 \end{pmatrix} \]Calculate the determinant of \( A \). Since \( A \) is an upper triangular matrix, the determinant is the product of its diagonal entries: \[ \text{det}(A) = 1 \times 1 \times 1 \times 1 = 1 eq 0 \]This indicates that the vectors are linearly independent and span \( \mathbb{R}^4 \).

Evaluate Set b

Given vectors: \((1,3,-5,0),(-2,1,0,0),(0,2,1,-1),(1,-4,5,0)\). Construct the matrix: \[ B = \begin{pmatrix} 1 & 3 & -5 & 0 \ -2 & 1 & 0 & 0 \ 0 & 2 & 1 & -1 \ 1 & -4 & 5 & 0 \end{pmatrix} \]Calculate the determinant of \( B \). Using row or column operations, simplify \( B \) to check for linear independence. After calculating, \[ \text{det}(B) = 0 \]As the determinant is zero, the vectors are linearly dependent and do not span \( \mathbb{R}^4 \).

Conclusion

For set (a), the vectors span \( \mathbb{R}^4 \) since they are linearly independent. For set (b), the vectors do not span \( \mathbb{R}^4 \), because they are linearly dependent.

Key concepts

Spanning Sets

Spanning sets are crucial in linear algebra. They help us understand how vectors can cover or "span" an entire vector space. To say a set of vectors spans \( \mathbb{R}^n \), means we can create any vector in that space by combining the spanning vectors in just the right way. This involves using linear combinations.

• A linear combination of vectors involves multiplying each vector by a coefficient and adding them up.
• If a set of vectors spans a space, any vector in that space can be expressed in terms of these vectors.

Consider a set of vectors in \( \mathbb{R}^4\). If they span \( \mathbb{R}^4\), you can form any vector in \( \mathbb{R}^4\) using them. This means the set must have enough independent vectors to cover the whole space.

Linear Independence

Linear independence is all about ensuring that vectors are "independent" from each other. This means no vector in the set can be written as a combination of others.

• A set of vectors is linearly independent if the only solution to the equation involving their linear combination is the trivial solution (all coefficients are zero).
• For a set of vectors to span \( \mathbb{R}^n \), it must be linearly independent and have exactly \( n \) vectors.

In our context, if you can say none of the vectors can be derived from others, they are independent. This is crucial for covering the entire space without redundancy.

Determinants

Determinants are a handy tool for checking linear independence. They affect whether a set of vectors spans a space.

• The determinant of a square matrix formed by these vectors tells us about their independence.
• If the determinant is non-zero, the vectors are linearly independent.
• A zero determinant means linear dependence, so the vectors cannot span the space fully.

Calculating the determinant helps determine whether the vectors fill the space they are meant to cover. It's a simple but powerful calculation in linear algebra.

Vector Spaces

Vector spaces are like the playground for vectors. They have rules that vectors must follow.

• Vector spaces are made of vectors that can be added together and multiplied by scalars.
• They include familiar spaces like \( \mathbb{R}^2 \), \( \mathbb{R}^3 \), and even \( \mathbb{R}^4 \).
• Each vector space has dimensions, defined by the number of vectors in a spanning set for that space.

When we talk about spanning \( \mathbb{R}^4 \), we're discussing whether vectors can create all possible vectors in this four-dimensional world. Understanding these concepts gives you the big picture of how vectors interact in space.