# Linear Algebra: Operations with Vector Spaces

• looper
In summary, the conversation is discussing the operations of subspace addition and intersection in a vector space. It is asked to show that the operation of subspace addition has a zero element, is associative, and that the operation of intersection is also associative. The question further asks if there is an identity element for the operation of intersection. The conversation ends with confusion about the definitions and concepts being discussed.
looper

## Homework Statement

Let V be a vector space over k and S the set of all subspaces of V. Consider the operation of subspace addition in S. Show that there is a zero in S for this operation and that the operation is associative. Consider the operation of intersection in S. Show that this operation is associative. Is there an identity for this operation (i.e., there is an E existing in S such that A intersect E = A for all E in S)?

## The Attempt at a Solution

Let U and W be in S. Then U+W=W+U=0+0. U intersect W = W intersect U...I'm not really sure where else to go with this...

What do you mean by "subspace addition"? The direct product? You say "U+ W= W+ U= 0+ 0". What is "0"?

HallsofIvy said:
What do you mean by "subspace addition"? The direct product? You say "U+ W= W+ U= 0+ 0". What is "0"?

I guess the 0 that E U and the 0 that E W? You know, I'm not really sure. We're not using a published book for my Linear Algebra class. We have pdf pages of the professor's lecture notes but they are really short (very minimal explanations), full of typos, and examples without solutions. To be honest, I'm not really sure even what the question is asking. =/

## 1. What is Linear Algebra?

Linear Algebra is a branch of mathematics that deals with operations and transformations on vector spaces, which are sets of objects that can be added together and multiplied by scalars. It is also used to solve systems of linear equations and study geometric concepts such as lines, planes, and hyperplanes.

## 2. What are vector spaces?

Vector spaces are sets of objects, called vectors, that can be added together and multiplied by scalars. They follow certain properties, such as closure under addition and scalar multiplication, and contain a zero vector and additive inverses. Examples of vector spaces include the set of real numbers, the set of polynomials, and the set of matrices.

## 3. What are the basic operations in Linear Algebra?

The basic operations in Linear Algebra include vector addition, scalar multiplication, and matrix multiplication. Vector addition involves adding two vectors together component-wise, while scalar multiplication involves multiplying a vector by a scalar value. Matrix multiplication involves multiplying two matrices together according to specific rules.

## 4. How is Linear Algebra used in real life?

Linear Algebra has many real-life applications, such as data analysis, computer graphics, and engineering. It is used to solve systems of linear equations, which can be used to model real-life situations. It is also used in machine learning and artificial intelligence for tasks such as image and speech recognition.

## 5. What are some important concepts in Linear Algebra?

Some important concepts in Linear Algebra include vector spaces, linear transformations, eigenvalues and eigenvectors, and orthogonality. Vector spaces are sets of objects that can be added and multiplied by scalars, while linear transformations are operations that preserve the structure of vector spaces. Eigenvalues and eigenvectors are important for understanding how a transformation affects a vector, and orthogonality is used to study perpendicularity and projection in vector spaces.

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