Linear Algebra, spanned by vectors

In summary, linear algebra is a branch of mathematics that deals with linear equations and their representations in vector spaces. It involves the use of matrices, vectors, and other mathematical structures to solve problems related to systems of linear equations, geometric transformations, and eigenvalues and eigenvectors. A set of vectors is said to span a space if every vector in that space can be written as a linear combination of those vectors. Vectors play a crucial role in linear algebra as they represent quantities with both magnitude and direction and are used to solve problems in various fields such as physics, engineering, and computer science. Linear algebra is also used in real-world applications such as optimization, data analysis, computer graphics, and machine learning in fields like robotics, image processing
  • #1
rocomath
1,755
1
How is this "All of R^3"

Describe the subspace of R^3 spanned by: all vectors with positive components.

Answer is, All of R^3. I don't get how it's all of R^3 though because if the components are all positive, it's only spanning in the positive direction? What about the negative portion? That is excluded, so it's not all of R^3.
 
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  • #2
That may be so, but the linear coefficients of the positive vectors may be negative.
 
  • #3
Defennder said:
That may be so, but the linear coefficients of the positive vectors may be negative.
Ah, I c! Thanks.
 

FAQ: Linear Algebra, spanned by vectors

What is linear algebra?

Linear algebra is a branch of mathematics that deals with the study of linear equations and their representations in vector spaces. It involves the use of matrices, vectors, and other mathematical structures to solve problems related to systems of linear equations, geometric transformations, and eigenvalues and eigenvectors.

What does it mean for a vector to span a space?

A set of vectors is said to span a space if every vector in that space can be written as a linear combination of those vectors. In other words, the span of a set of vectors is the set of all possible linear combinations of those vectors.

How do you determine if a set of vectors spans a space?

To determine if a set of vectors spans a space, you can use the row reduction method to convert the vectors into a matrix and check for linear independence. If the vectors are linearly independent, then they span the space. Another method is to check if the dimension of the space is equal to the number of vectors in the set.

What is the importance of vectors in linear algebra?

Vectors play a crucial role in linear algebra as they represent quantities that have both magnitude and direction. They are used to represent linear transformations and equations, and their properties are used to solve problems related to vector spaces and matrices.

How is linear algebra used in real-world applications?

Linear algebra has a wide range of applications in various fields such as physics, engineering, computer science, and economics. It is used to solve problems related to optimization, data analysis, computer graphics, and machine learning. It also has applications in robotics, image processing, and financial modeling.

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