Introduction to Vetors

In this note,i will make some summaries about vectors

Vectors and Linear Combination

We often write the vector as (1,2) for example,but in linear algebra,we try to define it as \(\begin{bmatrix} 1\\ 2\end{bmatrix}\)

1. A vector v in two-dimensional space has two components \(v_1\) and \(v_2\)
   ,when in three-dimensional will have three components

2. the $c v +d w $ is the linear combination of the vectors \(v\) and \(w\) (\(c,d\in R\))
   A linear combination of three vectors u and v and w is cu+dv+ew

3. Vectors obey the parallelogram law

4. The linear combinations of vectors can span a space

Length and Dot Products

1. The dot product \(v·w\) multiplies each component \(v_i\) by \(w_i\) and adds all \(v_iw_i\)

2. The length \(|v|\) is the square root of \(v·v\) .Then \(u=\frac{v}{|v|}\) is unit vector:length 1.

   • for example:The length of\(v=\begin{bmatrix} v_1\\ v_2\\..\\v_n\end{bmatrix}\)is \(\displaystyle \sqrt(\sum_{i=1}^{n}(v_i)^2)\)

3. The dot product is \(v·w=0\) when vectors v and w are perpendicular.

4. The cosine of \(\theta\)(the angle between any nonzero v and w) never exceeds 1:

• Cosine \(\cos\theta=\frac{v·w}{|v||w|}\)
• \(|v·w|<=|v||w|\)

Matrices

1. Matrix: A matrix is a higher-dimensional structure composed of multiple vectors. A ( m \(\times\) n ) matrix can be viewed as consisting of ( m ) column vectors or ( n ) row vectors.

2. Matrix times vector : Ax=combination of the columns of A.

3. The solution to \(Ax=b\) is x=\(A^{-1}b\),when A is an invertible matrix.

   • an invertible matrix is typically written as \(A^{-1}\).When $AA^{-1} = I $ holds, it indicates that the inverse \(A^{-1}\) exists.
   • In this context, \(I\) represents the identity matrix

4. A singular matrix is a square matrix that does not have an inverse.because Its columns lie in the same plane