## Proof that non-singular matrix A and singular matrix B add to non-singular matrix C

We have:     A note here is that where is not equivalent to . Consider the two respective column vectors and , for the sake of simplicity and are now scalars, we have but . Because is non-singular and suppose are by matrices, so in first columns are pivots and last columns are free. 10/24/2016: [...]

## [Notes] Reflection through hyperplane

(from Introduction to Linear Algebra [4th Edition] by Gilbert Strang, section 4.4, page 231) I don’t know the official reflection definition in mathematics so it would be a lame to prove that is indeed a reflection matrix, but I will note several stuffs that I have observed. Also special thanks to anyone gave me a [...]

## Proof that if a non-zero vector is orthogonal to a subspace then it is not in the subspace

We consider a non-zero vector and a matrix with set of column vectors is a basis , is orthogonal to . Now we prove that column vectors of are linearly independent. If column vectors of  are linearly independent then is invertible (which means ). We have: . It’s easy to see that  ( is invertible because has linearly [...]

## Proof that elimination does not change the row space

To be clear, in this proof I concern the row space of the matrix . The matrix represents the elimination process. We have:     is invertible (because it is the product of invertible elementary row operation matrices), so is . Hence we have:     That means column vectors of are linear combinations of column [...]

## Proof that dim(A) + dim(B) = dim(A and B) + dim(A + B)

Here and are two subspaces. For the simplicity, in the following proof, I will use notation as a matrix / block matrix containing basis of subspace , so will . The subspace contains all the vectors that satisfy the condition: There exists a vector  and vector  such that , which also means: . Every determines a [...]

## Proof that row operations do not change linear combination coefficents

Suppose I have: , and are vectors, are real numbers. If I do row operations on both sides by the matrix ( is in fact the notation of elementary matrix, however the matrix indicated here can do any complicated row operations) , I have to show that: It’s obvious that: . So the statement follows. The above statement [...]

## Pictorial proof that combinations of 3 independent vectors fill a 3-dimensional space

If three vectors (, , ) are independent then they will fill at least three planes ( from and , from  and , et cetera) (proof). These three planes will meet at the origin. For any arbitrary plane goes through the origin in the space. The plane cannot be parallel to any of the three above [...]

## Pictorial proof that combinations of two independent vectors fill a plane

Let’s take the intution like this: Imagine that from the intersectional point of  and , you draw an arbitrary vector . From the head of , we can always draw a vector such that it’s parallel to the line containing vector , the head of will touch the head of and the tail of will always [...]

## Learning From Data – A Short Course: Exercise B.1

Page 2: This exercise introduces some fundamental properties of vectors and bases. (a) Are the following sets of vectors dependent or independent? You can solve the equation system (with is a matrix of the set of vectors in consideration: ). If the equation system has no solution (except zero vector) then the set of vectors [...]