Expressing a Vector as a Linear Combination

Oct 13, 2025 by windailys.com 45 views

Show that $$R^2 = \operatorname{span}\left(\left[\begin{array}{r}3 \ -2\end{array}\right],\left[\begin{array}{l}0 \ 1\end{array}\right]\right)$$

In linear algebra, the span of a set of vectors is the set of all linear combinations of those vectors. In this article, we will show that the set of all linear combinations of the vectors $\left[\begin{array}{r}3 \\ -2\end{array}\right]$ and $\left[\begin{array}{l}0 \\ 1\end{array}\right]$ is equal to the entire 2-dimensional space $R^2$ .

Understanding the Problem

To show that the span of the given vectors is equal to $R^2$ , we need to demonstrate that any vector in $R^2$ can be expressed as a linear combination of the given vectors. In other words, we need to show that for any vector $\left[\begin{array}{r}x \\ y\end{array}\right]$ in $R^2$ , there exist scalars $c_1$ and $c_2$ such that:

\left[\begin{array}{r}x \\ y\end{array}\right] = c_1\left[\begin{array}{r}3 \\ -2\end{array}\right] + c_2\left[\begin{array}{l}0 \\ 1\end{array}\right]

Expressing a Vector as a Linear Combination

Let $\left[\begin{array}{r}x \\ y\end{array}\right]$ be any vector in $R^2$ . We can express this vector as a linear combination of the given vectors by solving the following system of equations:

\left[\begin{array}{r}3c_1 \\ -2c_1\end{array}\right] + \left[\begin{array}{l}0 \\ c_2\end{array}\right] = \left[\begin{array}{r}x \\ y\end{array}\right]

This system of equations can be written in matrix form as:

\left[\begin{array}{cc}3 & 0 \\ -2 & 1\end{array}\right]\left[\begin{array}{r}c_1 \\ c_2\end{array}\right] = \left[\begin{array}{r}x \\ y\end{array}\right]

Solving the System of Equations

To solve this system of equations, we can use the inverse of the coefficient matrix. The inverse of the coefficient matrix is given by:

\left[\begin{array}{cc}3 & 0 \\ -2 & 1\end{array}\right]^{-1} = \left[\begin{array}{cc}\frac{1}{3} & 0 \\ \frac{2}{3} & 1\end{array}\right]

Multiplying both sides of the equation by the inverse of the coefficient matrix, we get:

\left[\begin{array}{r}c_1 \\ c_2\end{array}\right] = \left[\begin{array}{cc}\frac{1}{3} & 0 \\ \frac{2}{3} & 1\end{array}\right]\left[\begin{array}{r}x \\ y\end{array}\right]

Finding the Values of $c_1$ and $c_2$

Multiplying the matrices, we get:

c_1 = \frac{1}{3}x

c_2 = \frac{2}{3}x + y

Conclusion

We have shown that for any vector $\left[\begin{array}{r}x \\ y\end{array}\right]$ in $R^2$ , there exist scalars $c_1$ and $c_2$ such that:

\left[\begin{array}{r}x \\ y\end{array}\right] = c_1\left[\begin{array}{r}3 \\ -2\end{array}\right] + c_2\left[\begin{array}{l}0 \\ 1\end{array}\right]

This demonstrates that the span of the vectors $\left[\begin{array}{r}3 \\ -2\end{array}\right]$ and $\left[\begin{array}{l}0 \\ 1\end{array}\right]$ is equal to the entire 2-dimensional space $R^2$ .

The Importance of Span

The concept of span is important in linear algebra because it allows us to describe the set of all linear combinations of a set of vectors. This is useful in a variety of applications, including solving systems of linear equations and finding the solution to a linear system.

The Relationship Between Span and Linear Independence

The span of a set of vectors is related to the concept of linear independence. A set of vectors is said to be linearly independent if none of the vectors in the set can be expressed as a linear combination of the other vectors. If a set of vectors is linearly independent, then the span of the set is equal to the entire space.

The Relationship Between Span and Basis

The span of a set of vectors is also related to the concept of a basis. A basis for a vector space is a set of vectors that spans the space and is linearly independent. If a set of vectors is a basis for a vector space, then the span of the set is equal to the entire space.

Conclusion

Q: What is the span of a set of vectors?

A: The span of a set of vectors is the set of all linear combinations of those vectors. In other words, it is the set of all vectors that can be expressed as a linear combination of the given vectors.

Q: How do you find the span of a set of vectors?

A: To find the span of a set of vectors, you need to find the set of all linear combinations of the given vectors. This can be done by solving a system of linear equations, where the coefficients of the linear combination are the unknowns.

Q: What is the difference between span and linear independence?

A: The span of a set of vectors is the set of all linear combinations of those vectors, while linear independence refers to the property that none of the vectors in the set can be expressed as a linear combination of the other vectors.

Q: What is the relationship between span and basis?

A: The span of a set of vectors is equal to the entire space if and only if the set is a basis for the space. A basis is a set of vectors that spans the space and is linearly independent.

Q: How do you determine if a set of vectors is a basis for a space?

A: To determine if a set of vectors is a basis for a space, you need to check two things: (1) that the set spans the space, and (2) that the set is linearly independent.

Q: What is the significance of the span of a set of vectors?

A: The span of a set of vectors is significant because it allows us to describe the set of all linear combinations of the given vectors. This is useful in a variety of applications, including solving systems of linear equations and finding the solution to a linear system.

Q: Can a set of vectors have multiple spans?

A: No, a set of vectors can only have one span. The span of a set of vectors is a unique set of vectors that includes all linear combinations of the given vectors.

Q: How do you find the span of a set of vectors in higher dimensions?

A: To find the span of a set of vectors in higher dimensions, you need to use a similar approach to the one used in two dimensions. However, the calculations can be more complex and may require the use of matrices and linear algebra techniques.

Q: What is the relationship between the span of a set of vectors and the concept of dimension?

A: The span of a set of vectors is related to the concept of dimension. The dimension of a space is the number of vectors in a basis for the space. The span of a set of vectors is equal to the entire space if and only if the set is a basis for the space, which means that the dimension of the space is equal to the number of vectors in the set.

Q: Can a set of vectors have a span that is not equal to the entire space?

A: Yes, a set of vectors can have a span that is not equal to the entire space. This occurs when the set is not a basis for the space, which means that the set does not span the entire space.

Q: How do you determine if a set of vectors has a span that is not equal to the entire space?

A: To determine if a set of vectors has a span that is not equal to the entire space, you need to check if the set is a basis for the space. If the set is not a basis for the space, then the span of the set is not equal to the entire space.

Conclusion

In this article, we have answered some common questions about the span of vectors, including what it is, how to find it, and its relationship to linear independence and basis. We have also discussed the significance of the span of a set of vectors and how it is related to the concept of dimension.