Definition questions for linear algebra

[futeca]

i am having trouble understanding some of the "basic" concepts of my linear algebra...any help would be greatly appreciated

what is an orthogonal basis? and how to construct it? i keep stumbling upon questions asking about construction a orthogonal basis for {v1, v2} in W

what i null A? need to understand the concept fro proofs

what is Col A? also need to understand for proofs

what exactly is a projection? i know the formula for it and i know how to apply it yet i don't understand the concept of what it is or how to picture it

thank you

 

[DonAntonio]

i am having trouble understanding some of the "basic" concepts of my linear algebra...any help would be greatly appreciated

what is an orthogonal basis? and how to construct it? i keep stumbling upon questions asking about construction a orthogonal basis for {v1, v2} in W

what i null A? need to understand the concept fro proofs

what is Col A? also need to understand for proofs

what exactly is a projection? i know the formula for it and i know how to apply it yet i don't understand the concept of what it is or how to picture it

thank you

Best advice ever: get yourself a good linear algebra book...:)

Ort. basis: read about Gram-Schmidt process and def. of orth. basis

Null A = most probably it means the null (sub)space of a matrix (or linear transformation) A, and it is the set of all vectors that A maps to the zero vector

Col A = probably it means the space spanned by the columns of an n x m matrix in the vector space [itex]\mathbb F^n[/itex]

DonAntonio

 

[futeca]

thank you so much this is great help!

btw do u happen to have any recommendations for a good linear algebra book to buy?

 

[Sankaku]

No need to buy. Here is a good one for free:

http://joshua.smcvt.edu/linearalgebra/

Here are some easier notes if that is too fast for you:

http://tutorial.math.lamar.edu/

 

[blue_raver22]

Linear algebra is a branch of mathematics that deals with the study of linear equations and their properties. It is a fundamental tool in many areas of science and engineering, and it has many applications in fields such as physics, computer science, and economics. In order to better understand the basic concepts of linear algebra, it is important to have a clear understanding of some key definitions.

An orthogonal basis is a set of vectors that are all mutually perpendicular to each other. This means that the dot product of any two vectors in the basis is equal to zero. To construct an orthogonal basis, one can use the Gram-Schmidt process, which involves taking a set of linearly independent vectors and finding a set of orthogonal vectors that span the same subspace.

The null space of a matrix A, denoted as null(A), is the set of all vectors x that when multiplied by A result in the zero vector. In other words, it is the set of solutions to the homogeneous equation Ax = 0. Understanding the concept of the null space is important in proofs involving linear transformations and systems of linear equations.

The column space of a matrix A, denoted as col(A), is the set of all linear combinations of the columns of A. In other words, it is the span of the columns of A. Similar to the null space, understanding the concept of the column space is important in proofs involving linear transformations and systems of linear equations.

A projection is a transformation that maps a vector onto a subspace in a way that preserves its length and direction. In other words, it "projects" the vector onto the subspace. The formula for a projection involves finding the dot product of the vector with the unit vectors in the subspace. Understanding the concept of a projection is important in applications such as least squares regression and finding the best fit line for a set of data points.

I hope this explanation helps you better understand the basic concepts of linear algebra. It is important to have a clear understanding of these definitions in order to fully grasp the applications and proofs in this field of mathematics. If you continue to have trouble, I suggest seeking additional resources such as textbooks or online tutorials to further your understanding.